37.7 Elliptic curves over the rational numbers

Module: sage.schemes.elliptic_curves.ell_rational_field

Elliptic curves over the rational numbers

Author Log:

Module-level Functions

cremona_curves( conductors)

Return iterator over all known curves (in database) with conductor in the list of conductors.

sage: [(E.label(), E.rank()) for E in cremona_curves(srange(35,40))]
[('35a1', 0),
('35a2', 0),
('35a3', 0),
('36a1', 0),
('36a2', 0),
('36a3', 0),
('36a4', 0),
('37a1', 1),
('37b1', 0),
('37b2', 0),
('37b3', 0),
('38a1', 0),
('38a2', 0),
('38a3', 0),
('38b1', 0),
('38b2', 0),
('39a1', 0),
('39a2', 0),
('39a3', 0),
('39a4', 0)]

cremona_optimal_curves( conductors)

Return iterator over all known optimal curves (in database) with conductor in the list of conductors.

sage: [(E.label(), E.rank()) for E in cremona_optimal_curves(srange(35,40))]
[('35a1', 0),
('36a1', 0),
('37a1', 1),
('37b1', 0),
('38a1', 0),
('38b1', 0),
('39a1', 0)]

Class: EllipticCurve_rational_field

class EllipticCurve_rational_field
Elliptic curve over the Rational Field.
EllipticCurve_rational_field( self, ainvs, [extra=None])

Functions: an,$ \,$ analytic_rank,$ \,$ anlist,$ \,$ ap,$ \,$ aplist,$ \,$ cm_discriminant,$ \,$ conductor,$ \,$ CPS_height_bound,$ \,$ cremona_label,$ \,$ database_curve,$ \,$ eval_modular_form,$ \,$ gens,$ \,$ gens_certain,$ \,$ global_integral_model,$ \,$ has_cm,$ \,$ heegner_discriminants,$ \,$ heegner_discriminants_list,$ \,$ heegner_index,$ \,$ heegner_index_bound,$ \,$ heegner_point_height,$ \,$ integral_model,$ \,$ integral_weierstrass_model,$ \,$ is_global_integral_model,$ \,$ is_good,$ \,$ is_integral,$ \,$ is_irreducible,$ \,$ is_local_integral_model,$ \,$ is_minimal,$ \,$ is_ordinary,$ \,$ is_reducible,$ \,$ is_semistable,$ \,$ is_supersingular,$ \,$ is_surjective,$ \,$ isogeny_class,$ \,$ isogeny_graph,$ \,$ kodaira_type,$ \,$ label,$ \,$ Lambda,$ \,$ local_integral_model,$ \,$ lseries,$ \,$ matrix_of_frobenius,$ \,$ minimal_model,$ \,$ mod5family,$ \,$ modular_degree,$ \,$ modular_form,$ \,$ modular_parametrization,$ \,$ modular_symbol,$ \,$ modular_symbol_space,$ \,$ mwrank,$ \,$ mwrank_curve,$ \,$ newform,$ \,$ ngens,$ \,$ non_surjective,$ \,$ Np,$ \,$ ordinary_primes,$ \,$ p_isogenous_curves,$ \,$ padic_E2,$ \,$ padic_height,$ \,$ padic_height_pairing_matrix,$ \,$ padic_height_via_multiply,$ \,$ padic_lseries,$ \,$ padic_regulator,$ \,$ padic_sigma,$ \,$ padic_sigma_truncated,$ \,$ pari_curve,$ \,$ pari_mincurve,$ \,$ period_lattice,$ \,$ point_search,$ \,$ q_eigenform,$ \,$ q_expansion,$ \,$ quadratic_twist,$ \,$ rank,$ \,$ real_components,$ \,$ reducible_primes,$ \,$ regulator,$ \,$ root_number,$ \,$ satisfies_heegner_hypothesis,$ \,$ saturation,$ \,$ sea,$ \,$ selmer_rank_bound,$ \,$ sha,$ \,$ silverman_height_bound,$ \,$ simon_two_descent,$ \,$ supersingular_primes,$ \,$ tamagawa_number,$ \,$ tamagawa_numbers,$ \,$ tamagawa_product,$ \,$ tate_curve,$ \,$ three_selmer_rank,$ \,$ torsion_order,$ \,$ torsion_subgroup,$ \,$ two_descent,$ \,$ two_descent_simon,$ \,$ two_torsion_rank

an( self, n)

The n-th Fourier coefficient of the modular form corresponding to this elliptic curve, where n is a positive integer.

sage: E=EllipticCurve('37a1')
sage: [E.an(n) for n in range(20) if n>0]
[1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0]

analytic_rank( self, [algorithm=cremona])

Return an integer that is probably the analytic rank of this elliptic curve.

Input: algorithm:

- 'cremona' (default) - Use the Buhler-Gross algorithm as implemented in GP by Tom Womack and John Cremona, who note that their implementation is practical for any rank and conductor $ \leq 10^{10}$ in 10 minutes.

- 'sympow' -use Watkins's program sympow

- 'rubinstein' - use Rubinstein's L-function C++ program lcalc.

- 'magma' - use MAGMA

- 'all' - compute with all other free algorithms, check that the answers agree, and return the common answer.

Note: If the curve is loaded from the large Cremona database, then the modular degree is taken from the database.

Of the three above, probably Rubinstein's is the most efficient (in some limited testing I've done).

Note: It is an open problem to prove that any particular elliptic curve has analytic rank $ \geq 4$ .

sage: E = EllipticCurve('389a')
sage: E.analytic_rank(algorithm='cremona')
2
sage: E.analytic_rank(algorithm='rubinstein')
2
sage: E.analytic_rank(algorithm='sympow')
2
sage: E.analytic_rank(algorithm='magma')    # optional
2
sage: E.analytic_rank(algorithm='all')
2

anlist( self, n, [python_ints=False])

The Fourier coefficients up to and including $ a_n$ of the modular form attached to this elliptic curve. The ith element of the return list is a[i].

Input:

n
- integer
python_ints
- bool (default: False); if True return a list of Python ints instead of SAGE integers.

Output:
- list of integers

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.anlist(3)
[0, 1, -2, -1]

sage: E = EllipticCurve([0,1])
sage: E.anlist(20)
[0, 1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 8, 0]

ap( self, p)

The p-th Fourier coefficient of the modular form corresponding to this elliptic curve, where p is prime.

sage: E=EllipticCurve('37a1')
sage: [E.ap(p) for p in prime_range(50)]
[-2, -3, -2, -1, -5, -2, 0, 0, 2, 6, -4, -1, -9, 2, -9]

aplist( self, n, [python_ints=False])

The Fourier coefficients $ a_p$ of the modular form attached to this elliptic curve, for all primes $ p\leq n$ .

Input:

n
- integer
python_ints
- bool (default: False); if True return a list of Python ints instead of SAGE integers.

Output:
- list of integers

sage: e = EllipticCurve('37a')
sage: e.aplist(1)
[]
sage: e.aplist(2)
[-2]
sage: e.aplist(10)
[-2, -3, -2, -1]
sage: v = e.aplist(13); v
[-2, -3, -2, -1, -5, -2]
sage: type(v[0])
<type 'sage.rings.integer.Integer'>
sage: type(e.aplist(13, python_ints=True)[0])
<type 'int'>

cm_discriminant( self)

Returns the associated quadratic discriminant if this elliptic curve has Complex Multiplication.

A ValueError is raised if the curve does not have CM (see the function has_cm())

 sage: E=EllipticCurve('32a1')
 sage: E.cm_discriminant()
 -4
 sage: E=EllipticCurve('121b1')
 sage: E.cm_discriminant()
 -11
 sage: E=EllipticCurve('37a1')
sage: E.cm_discriminant()
Traceback (most recent call last):
...
ValueError: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
does not have CM

conductor( self, [algorithm=pari])

Returns the conductor of the elliptic curve.

Input:

algorithm
- str, (default: "pari")
"pari"
- use the PARI C-library ellglobalred implementation of Tate's algorithm
"mwrank"
- use Cremona's mwrank implementation of Tate's algorithm; can be faster if the curve has integer coefficients (TODO: limited to small conductor until mwrank gets integer factorization)
"gp"
- use the GP interpreter.
"all"
- use both implementations, verify that the results are the same (or raise an error), and output the common value.

sage: E = EllipticCurve([1, -1, 1, -29372, -1932937])
sage: E.conductor(algorithm="pari")
3006
sage: E.conductor(algorithm="mwrank")
3006
sage: E.conductor(algorithm="gp")
3006
sage: E.conductor(algorithm="all")
3006

NOTE: The conductor computed using each algorithm is cached separately. Thus calling E.conductor("pari"), then E.conductor("mwrank") and getting the same result checks that both systems compute the same answer.

CPS_height_bound( self)

Return the Cremona-Prickett-Siksek height bound. This is a floating point number B such that if P is a point on the curve, then the naive logarithmetic height of P is off from the canonical height by at most B.

sage: E = EllipticCurve("11a")
sage: E.CPS_height_bound()
2.8774743273580445
sage: E = EllipticCurve("5077a")
sage: E.CPS_height_bound()
0.0
sage: E = EllipticCurve([1,2,3,4,1])
sage: E.CPS_height_bound()
Traceback (most recent call last):
...
RuntimeError: curve must be minimal.
sage: F = E.quadratic_twist(-19)
sage: F
Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 + 1376*x - 130 over
Rational Field
sage: F.CPS_height_bound()
0.65551583769728516

IMPLEMENTATION: Call the corresponding mwrank C++ library function.

cremona_label( self, [space=False])

Return the Cremona label associated to (the minimal model) of this curve, if it is known. If not, raise a RuntimeError exception.

sage: E=EllipticCurve('389a1')
sage: E.cremona_label()
'389a1'

The default database only contains conductors up to 10000, so any curve with conductor greater than that will cause an error to be raised:

sage: E=EllipticCurve([1,2,3,4,5]);
sage: E.conductor()
10351
sage: E.cremona_label()
Traceback (most recent call last):
...
RuntimeError: Cremona label not known for Elliptic Curve defined by y^2 +
x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field.

database_curve( self)

Return the curve in the elliptic curve database isomorphic to this curve, if possible. Otherwise raise a RuntimeError exception.

sage: E = EllipticCurve([0,1,2,3,4])
sage: E.database_curve()
Elliptic Curve defined by y^2  = x^3 + x^2 + 3*x + 5 over Rational Field

NOTES: The model of the curve in the database can be different than the Weierstrass model for this curve, e.g., database models are always minimal.

eval_modular_form( self, points, prec)

Evaluate the L-series of this elliptic curve at points in CC

Input: points- a list of points in the half-plane of convergence prec- precision

Output: A list of values L(E,s) for s in points

NOTE: Better examples are welcome.

sage: E=EllipticCurve('37a1')
sage: E.eval_modular_form([1.5+I,2.0+I,2.5+I],0.000001)
[0, 0, 0]

gens( self, [verbose=False], [rank1_search=10], [algorithm=mwrank_shell], [only_use_mwrank=True], [proof=None])

Compute and return generators for the Mordell-Weil group E(Q) *modulo* torsion.

HINT: If you would like to control the height bounds used in the 2-descent, first call the two_descent function with those height bounds. However that function, while it displays a lot of output, returns no values.

TODO: (1) Right now this function assumes that the input curve is in minimal Weierstrass form. This restriction will be removed in the future. This function raises a NotImplementedError if a non-minimal curve is given as input.

(2) Allow passing of command-line parameters to mwrank.

WARNING: If the program fails to give a provably correct result, it prints a warning message, but does not raise an exception. Use the gens_certain command to find out if this warning message was printed.

Input:

verbose
- (default: None), if specified changes the verbosity of mwrank computations.
rank1_search
- (default: 16), if the curve has analytic rank 1, try to find a generator by a direct search up to this logarithmic height. If this fails the usual mwrank procedure is called.
algorithm
- 'mwrank_shell' (default) - call mwrank shell command
- 'mwrank_lib' - call mwrank c library
only_use_mwrank
- bool (default True) if false, attempts to first use more naive, natively implemented methods.
proof
- bool or None (default None, see proof.elliptic_curve or sage.structure.proof).
Output:
generators
- List of generators for the Mordell-Weil group.

IMPLEMENTATION: Uses Cremona's mwrank C library.

sage: E = EllipticCurve('389a')
sage: E.gens()                 # random output    
[(-1 : 1 : 1), (0 : 0 : 1)]

gens_certain( self)

Return True if the generators have been proven correct.

sage: E=EllipticCurve('37a1')
sage: E.gens()
[(0 : -1 : 1)]
sage: E.gens_certain()
True

global_integral_model( self)

Return a model of self which is integral at all primes

sage: E = EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
sage: F = E.global_integral_model(); F
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: F == EllipticCurve('5077a1')
True

has_cm( self)

Returns True iff this elliptic curve has Complex Multiplication

sage: E=EllipticCurve('37a1')
sage: E.has_cm()
False
sage: E=EllipticCurve('32a1')
sage: E.has_cm()
True
sage: E.j_invariant()
1728

heegner_discriminants( self, bound)

Return the list of self's Heegner discriminants between -1 and -bound.

Input:

bound (int)
- upper bound for -discriminant

Output: The list of Heegner discriminants between -1 and -bound for the given elliptic curve.

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants(30)
[-7, -8, -19, -24]

heegner_discriminants_list( self, n)

Return the list of self's first n Heegner discriminants smaller than -5.

Input:

n (int)
- the number of discriminants to compute

Output: The list of the first n Heegner discriminants smaller than -5 for the given elliptic curve.

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants_list(4)
[-7, -8, -19, -24]

heegner_index( self, D, [min_p=2], [prec=5], [verbose=False])

Return an interval that contains the index of the Heegner point $ y_K$ in the group of K-rational points modulo torsion on this elliptic curve, computed using the Gross-Zagier formula and/or a point search, or the index divided by $ 2$ .

NOTES: If min_p is bigger than 2 then the index can be off by any prime less than min_p. This function returns the index divided by $ 2$ exactly when $ E(\mathbf{Q})_{/tor}$ has index $ 2$ in $ E(K)_{/tor}$ .

Input:

D (int)
- Heegner discriminant
min_p (int)
- (default: 2) only rule out primes >= min_p dividing the index.
verbose (bool)
- (default: False); print lots of mwrank search status information when computing regulator
prec (int)
- (default: 5), use prec*sqrt(N) + 20 terms of L-series in computations, where N is the conductor.

Output: an interval that contains the index

sage: E = EllipticCurve('11a')
sage: E.heegner_discriminants(50)
[-7, -8, -19, -24, -35, -39, -40, -43]
sage: E.heegner_index(-7)
[0.99999332 .. 1.0000077]

sage: E = EllipticCurve('37b')
sage: E.heegner_discriminants(100)
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
sage: E.heegner_index(-95)          # long time (1 second)
[1.9999923 .. 2.0000077]

Current discriminants -3 and -4 are not supported:

sage: E.heegner_index(-3)
Traceback (most recent call last):
...
ArithmeticError: Discriminant (=-3) must not be -3 or -4.

The curve 681b returns an interval that contains $ 3/2$ . This is because $ E(\mathbf{Q})$ is not saturated in $ E(K)$ . The true index is $ 3$ :

sage: E = EllipticCurve('681b')
sage: I = E.heegner_index(-8); I
[1.4999942 .. 1.5000058]
sage: 2*I
[2.9999885 .. 3.0000115]

In fact, whenever the returned index has a denominator of $ 2$ , the true index is got by multiplying the returned index by $ 2$ . Unfortunately, this is not an if and only if condition, i.e., sometimes the index must be multiplied by $ 2$ even though the denominator is not $ 2$ .

heegner_index_bound( self, [D=0], [prec=5], [verbose=True], [max_height=21])

Assume self has rank 0.

Return a list v of primes such that if an odd prime p divides the index of the Heegner point in the group of rational points *modulo torsion*, then p is in v.

If 0 is in the interval of the height of the Heegner point computed to the given prec, then this function returns v = 0. This does not mean that the Heegner point is torsion, just that it is very likely torsion.

If we obtain no information from a search up to max_height, e.g., if the Siksek et al. bound is bigger than max_height, then we return v = -1.

Input:

D (int)
- (deault: 0) Heegner discriminant; if 0, use the first discriminant < -4 that satisfies the Heegner hypothesis
verbose (bool)
- (default: True)
prec (int)
- (default: 5), use prec*sqrt(N) + 20 terms of L-series in computations, where N is the conductor.
max_height (float)
- should be <= 21; bound on logarithmic naive height used in point searches. Make smaller to make this function faster, at the expense of possibly obtaining a worse answer. A good range is between 13 and 21.

Output:
v
- list or int (bad primes or 0 or -1)
D
- the discriminant that was used (this is useful if D was automatically selected).

sage: E = EllipticCurve('11a1')
sage: E.heegner_index_bound(verbose=False)
([2], -7)

heegner_point_height( self, D, [prec=2])

Use the Gross-Zagier formula to compute the Neron-Tate canonical height over K of the Heegner point corresponding to D, as an Interval (since it's computed to some precision using L-functions).

Input:

D (int)
- fundamental discriminant (=/= -3, -4)
prec (int)
- (default: 2), use prec*sqrt(N) + 20 terms of L-series in computations, where N is the conductor.

Output: Interval that contains the height of the Heegner point.

sage: E = EllipticCurve('11a')
sage: E.heegner_point_height(-7)
[0.22226977 ... 0.22227479]

integral_model( self)

Return a weierstrass model, $ F$ , of self with integral coefficients, along with a morphism $ \phi$ of points on self to points on $ F$ .

sage: E = EllipticCurve([1/2,0,0,5,1/3])
sage: F, phi = E.integral_model()
sage: F
Elliptic Curve defined by y^2 + 3*x*y  = x^3 + 6480*x + 15552 over Rational
Field
sage: phi
Generic morphism:
  From: Abelian group of points on Elliptic Curve defined by y^2 + 1/2*x*y 
= x^3 + 5*x + 1/3 over Rational Field
  To:   Abelian group of points on Elliptic Curve defined by y^2 + 3*x*y  =
x^3 + 6480*x + 15552 over Rational Field
  Via:  (u,r,s,t) = (1/6, 0, 0, 0)
sage: P = E([4/9,41/27])
sage: phi(P)
(16 : 328 : 1)
sage: phi(P) in F
True

integral_weierstrass_model( self)

Return a model of the form $ y^2 = x^3 + a*x + b$ for this curve with $ a,b\in\mathbf{Z}$ .

sage: E = EllipticCurve('17a1')
sage: E.integral_weierstrass_model()
Elliptic Curve defined by y^2  = x^3 - 11*x - 890 over Rational Field

is_global_integral_model( self)

Return true iff self is integral at all primes

sage: E=EllipticCurve([1/2,1/5,1/5,1/5,1/5])
sage: E.is_global_integral_model()
False
sage: Emin=E.global_integral_model()
sage: Emin.is_global_integral_model()
True

is_good( self, p, [check=True])

Return True if $ p$ is a prime of good reduction for $ E$ .

Input:

p
- a prime

Output: bool

sage: e = EllipticCurve('11a')
sage: e.is_good(-8)
Traceback (most recent call last):
...
ValueError: p must be prime
sage: e.is_good(-8, check=False)
True

is_integral( self)

Returns True if this elliptic curve has integral coefficients (in Z)

sage: E=EllipticCurve(QQ,[1,1]); E
Elliptic Curve defined by y^2  = x^3 + x +1 over Rational Field
sage: E.is_integral()
True
sage: E2=E.change_weierstrass_model(2,0,0,0); E2
Elliptic Curve defined by y^2  = x^3 + 1/16*x + 1/64 over Rational Field
sage: E2.is_integral()
False

is_irreducible( self, p)

Return True if the mod p represenation is irreducible.

sage: e = EllipticCurve('37b')
sage: e.is_irreducible(2)
True
sage: e.is_irreducible(3)
False
sage: e.is_reducible(2)
False
sage: e.is_reducible(3)
True

is_local_integral_model( self)

Tests if self is integral at the prime $ p$ , or at all the primes if $ p$ is a list or tuple of primes

sage: E=EllipticCurve([1/2,1/5,1/5,1/5,1/5])
sage: [E.is_local_integral_model(p) for p in (2,3,5)]
[False, True, False]
sage: E.is_local_integral_model(2,3,5)
False
sage: Eint2=E.local_integral_model(2)
sage: Eint2.is_local_integral_model(2)
True

is_minimal( self)

Return True iff this elliptic curve is a reduced minimal model.

the unique minimal Weierstrass equation for this elliptic curve. This is the model with minimal discriminant and $ a_1,a_2,a_3 \in \{0,\pm 1\}$ .

TO DO: This is not very efficient since it just computes the minimal model and compares. A better implementation using the Kraus conditions would be preferable.

sage: E=EllipticCurve([10,100,1000,10000,1000000])
sage: E.is_minimal()
False
sage: E=E.minimal_model()
sage: E.is_minimal()
True

is_ordinary( self, p, [ell=None])

Return True precisely when the mod-p representation attached to this elliptic curve is ordinary at ell.

Input:

p
- a prime ell - a prime (default: p)

Output: bool

sage: E=EllipticCurve('37a1')
sage: E.is_ordinary(37)
True
sage: E=EllipticCurve('32a1')
sage: E.is_ordinary(2)
False
sage: [p for p in prime_range(50) if E.is_ordinary(p)]
[5, 13, 17, 29, 37, 41]

is_reducible( self, p)

Return True if the mod-p representation attached to E is reducible.

Input:

p
- a prime number

NOTE: The answer is cached.

sage: E = EllipticCurve('121a'); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 30*x - 76 over
Rational Field
sage: E.is_reducible(7)
False
sage: E.is_reducible(11)
True
sage: EllipticCurve('11a').is_reducible(5)
True
sage: e = EllipticCurve('11a2')
sage: e.is_reducible(5)
True
sage: e.torsion_order()
1

is_semistable( self)

Return True iff this elliptic curve is semi-stable at all primes.

sage: E=EllipticCurve('37a1')
sage: E.is_semistable()
True
sage: E=EllipticCurve('90a1')
sage: E.is_semistable()
False

is_supersingular( self, p, [ell=None])

Return True precisely when p is a prime of good reduction and the mod-p representation attached to this elliptic curve is supersingular at ell.

Input:

p
- a prime ell - a prime (default: p)

Output: bool

sage: E=EllipticCurve('37a1')
sage: E.is_supersingular(37)
False
sage: E=EllipticCurve('32a1')
sage: E.is_supersingular(2)
False
sage: E.is_supersingular(7)
True
sage: [p for p in prime_range(50) if E.is_supersingular(p)]
[3, 7, 11, 19, 23, 31, 43, 47]

is_surjective( self, p, [A=1000])

Return True if the mod-p representation attached to E is surjective, False if it is not, or None if we were unable to determine whether it is or not.

NOTE: The answer is cached.

Input:

p
- int (a prime number)
A
- int (a bound on the number of a_p to use)

Output: a 2-tuple:
- surjective or (probably) not
- information about what it is if not surjective

sage: e = EllipticCurve('37b')
sage: e.is_surjective(2)
(True, None)
sage: e.is_surjective(3)
(False, '3-torsion')

REMARKS:

1. If p >= 5 then the mod-p representation is surjective if and only if the p-adic representation is surjective. When p = 2, 3 there are counterexamples. See a very recent paper of Elkies for more details when p=3.

2. When p <= 3 this function always gives the correct result irregardless of A, since it explicitly determines the p-division polynomial.

isogeny_class( self, [algorithm=mwrank], [verbose=False])

Return all curves over $ \mathbf{Q}$ in the isogeny class of this elliptic curve.

Input:

algorithm
- string:
"mwrank"
- (default) use the mwrank C++ library
"database"
- use the Cremona database (only works if curve is isomorphic to a curve in the database)

Output: Returns the sorted list of the curves isogenous to self. If algorithm is "mwrank", also returns the isogeny matrix (otherwise returns None as second return value).

Note: The result is not provably correct, in the sense that when the numbers are huge isogenies could be missed because of precision issues.

Note: The ordering depends on which algorithm is used.

sage: I, A = EllipticCurve('37b').isogeny_class('mwrank')  
sage: I   # randomly ordered 
[Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational
Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over
Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x +1 over Rational
Field]
sage: A
[0 3 3]
[3 0 0]
[3 0 0]

sage: I, _ = EllipticCurve('37b').isogeny_class('database'); I
[Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over
Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational
Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x +1 over Rational
Field]

This is an example of a curve with a $ 37$ -isogeny:

sage: E = EllipticCurve([1,1,1,-8,6])
sage: E.isogeny_class ()
([Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 8*x + 6 over
Rational Field,
  Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 208083*x - 36621194
over Rational Field],
 [ 0 37]
 [37  0])

This curve had numerous $ 2$ -isogenies:

sage: e=EllipticCurve([1,0,0,-39,90])
    sage: e.isogeny_class ()
    ([Elliptic Curve defined by y^2 + x*y  = x^3 - 39*x + 90 over Rational
Field,
      Elliptic Curve defined by y^2 + x*y  = x^3 - 4*x -1 over Rational
Field,
      Elliptic Curve defined by y^2 + x*y  = x^3 + x over Rational Field,
      Elliptic Curve defined by y^2 + x*y  = x^3 - 49*x - 136 over Rational
Field,
      Elliptic Curve defined by y^2 + x*y  = x^3 - 34*x - 217 over Rational
Field,
      Elliptic Curve defined by y^2 + x*y  = x^3 - 784*x - 8515 over
Rational Field],
     [0 2 0 0 0 0]
     [2 0 2 2 0 0]
     [0 2 0 0 0 0]
     [0 2 0 0 2 2]
     [0 0 0 2 0 0]
     [0 0 0 2 0 0])

See http://modular.ucsd.edu/Tables/nature/ for more interesting examples of isogeny structures.

isogeny_graph( self)

Returns a graph representing the isogeny class of this elliptic curve, where the vertices are isogenous curves over $ \mathbf{Q}$ and the edges are prime degree isogenies labeled by their degree.

sage: LL = []
sage: for e in cremona_optimal_curves(range(1, 38)):
...    G = e.isogeny_graph()
...    already = False
...    for H in LL:
...        if G.is_isomorphic(H):
...            already = True
...            break
...    if not already:
...        LL.append(G)
...
sage: graphs_list.show_graphs(LL)

sage: E = EllipticCurve('195a')
sage: G = E.isogeny_graph()
sage: for v in G: print v, G.get_vertex(v)
...
0 Elliptic Curve defined by y^2 + x*y  = x^3 - 110*x + 435 over Rational
Field
1 Elliptic Curve defined by y^2 + x*y  = x^3 - 115*x + 392 over Rational
Field
2 Elliptic Curve defined by y^2 + x*y  = x^3 + 210*x + 2277 over Rational
Field
3 Elliptic Curve defined by y^2 + x*y  = x^3 - 520*x - 4225 over Rational
Field
4 Elliptic Curve defined by y^2 + x*y  = x^3 + 605*x - 19750 over Rational
Field
5 Elliptic Curve defined by y^2 + x*y  = x^3 - 8125*x - 282568 over
Rational Field
6 Elliptic Curve defined by y^2 + x*y  = x^3 - 7930*x - 296725 over
Rational Field
7 Elliptic Curve defined by y^2 + x*y  = x^3 - 130000*x - 18051943 over
Rational Field
sage: G.plot(edge_labels=True).save('isogeny_graph.png')

kodaira_type( self, p)

Local Kodaira type of the elliptic curve at $ p$ .

Input:

- p, an integral prime
Output:
- the kodaira type of this elliptic curve at p, as a KodairaSymbol.

sage: E = EllipticCurve('124a')
sage: E.kodaira_type(2)
IV

label( self, [space=False])

Return the Cremona label associated to (the minimal model) of this curve, if it is known. If not, raise a RuntimeError exception.

sage: E=EllipticCurve('389a1')
sage: E.cremona_label()
'389a1'

The default database only contains conductors up to 10000, so any curve with conductor greater than that will cause an error to be raised:

sage: E=EllipticCurve([1,2,3,4,5]);
sage: E.conductor()
10351
sage: E.cremona_label()
Traceback (most recent call last):
...
RuntimeError: Cremona label not known for Elliptic Curve defined by y^2 +
x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field.

Lambda( self, s, prec)

Returns the value of the Lambda-series of the elliptic curve E at s, where s can be any complex number.

IMPLEMENTATION: Fairly *slow* computation using the definitions and implemented in Python.

Uses prec terms of the power series.

sage: E = EllipticCurve('389a')
sage: E.Lambda(1.4+0.5*I, 50)
-0.354172680517... + 0.874518681720...*I

local_integral_model( self, p)

Return a model of self which is integral at the prime $ p$

      sage: E=EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
      sage: E.local_integral_model(2)
Elliptic Curve defined by y^2 + 1/27*y = x^3 - 7/81*x + 2/243 over Rational
Field
      sage: E.local_integral_model(3)
       Elliptic Curve defined by y^2 + 1/8*y = x^3 - 7/16*x + 3/32 over
Rational Field
      sage: E.local_integral_model(2).local_integral_model(3) == EllipticCurve('5077a1')
      True

lseries( self)

Returns the L-series of this elliptic curve.

Further documentation is available for the functions which apply to the L-series.

sage: E=EllipticCurve('37a1')
sage: E.lseries()
Complex L-series of the Elliptic Curve defined by y^2 + y = x^3 - x over
Rational Field

matrix_of_frobenius( self, p, [prec=20], [check=False], [check_hypotheses=True], [algorithm=auto])

See the parameters and documentation for padic_E2.

minimal_model( self)

Return the unique minimal Weierstrass equation for this elliptic curve. This is the model with minimal discriminant and $ a_1,a_2,a_3 \in \{0,\pm 1\}$ .

sage: E=EllipticCurve([10,100,1000,10000,1000000])
sage: E.minimal_model()
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x +1 over Rational
Field

mod5family( self)

Return the family of all elliptic curves with the same mod-5 representation as self.

sage: E=EllipticCurve('32a1')
sage: E.mod5family()
Elliptic Curve defined by y^2  = x^3 + 4*x over Fraction Field of
Univariate Polynomial Ring in t over Rational Field

modular_degree( self, [algorithm=sympow])

Return the modular degree of this elliptic curve.

The result is cached. Subsequence calls, even with a different algorithm, just returned the cached result.

Input:

algorithm
- string:
'sympow'
- (default) use Mark Watkin's (newer) C program sympow
'magma'
- requires that MAGMA be installed (also implemented by Mark Watkins)

Note: On 64-bit computers ec does not work, so Sage uses sympow even if ec is selected on a 64-bit computer.

The correctness of this function when called with algorithm "ec" is subject to the following three hypothesis:

Moreover for all algorithms, computing a certain value of an $ L$ -function ``uses a heuristic method that discerns when the real-number approximation to the modular degree is within epsilon [=0.01 for algorithm="sympow"] of the same integer for 3 consecutive trials (which occur maybe every 25000 coefficients or so). Probably it could just round at some point. For rigour, you would need to bound the tail by assuming (essentially) that all the $ a_n$ are as large as possible, but in practise they exhibit significant (square root) cancellation. One difficulty is that it doesn't do the sum in 1-2-3-4 order; it uses 1-2-4-8--3-6-12-24-9-18- (Euler product style) instead, and so you have to guess ahead of time at what point to curtail this expansion.'' (Quote from an email of Mark Watkins.)

Note: If the curve is loaded from the large Cremona database, then the modular degree is taken from the database.

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational
Field
sage: E.modular_degree()
1                                      
sage: E = EllipticCurve('5077a')
sage: E.modular_degree()
1984                                   
sage: factor(1984)
2^6 * 31

sage: EllipticCurve([0, 0, 1, -7, 6]).modular_degree()
1984
sage: EllipticCurve([0, 0, 1, -7, 6]).modular_degree(algorithm='sympow')
1984
sage: EllipticCurve([0, 0, 1, -7, 6]).modular_degree(algorithm='magma')  # optional
1984

We compute the modular degree of the curve with rank 4 having smallest (known) conductor:

sage: E = EllipticCurve([1, -1, 0, -79, 289]) 
sage: factor(E.conductor())  # conductor is 234446
2 * 117223
sage: factor(E.modular_degree())
2^7 * 2617

modular_form( self)

Return the cuspidal modular form associated to this elliptic curve.

sage: E = EllipticCurve('37a')
sage: f = E.modular_form()
sage: f
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)

If you need to see more terms in the $ q$ -expansion:

sage: f.q_expansion(20)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - 5*q^11 -
6*q^12 - 2*q^13 + 2*q^14 + 6*q^15 - 4*q^16 - 12*q^18 + O(q^20)

NOTE: If you just want the $ q$ -expansion, use self.q_expansion(prec).

modular_parametrization( self)

Computes and returns the modular parametrization of this elliptic curve.

The curve is converted to a minimal model.

Output: a list of two larent series [x(x),y(x)] of degrees -2, -3 respectively, which satisfy the equation of the (minimal model of the) elliptic curve. The are modular functions on $ \Gamma_0(N)$ where $ N$ is the conductor.

X.deriv()/(2*Y+a1*X+a3) should equal f(q)dq/q where f is self.q_expansion().

sage: E=EllipticCurve('389a1')
sage: X,Y=E.modular_parametrization()
sage: X
q^-2 + 2*q^-1 + 4 + 7*q + 13*q^2 + 18*q^3 + 31*q^4 + 49*q^5 + 74*q^6 +
111*q^7 + 173*q^8 + 251*q^9 + 379*q^10 + 560*q^11 + 824*q^12 + 1199*q^13 +
1773*q^14 + 2365*q^15 + 3463*q^16 + 4508*q^17 + O(q^18)
sage: Y
-q^-3 - 3*q^-2 - 8*q^-1 - 17 - 33*q - 61*q^2 - 110*q^3 - 186*q^4 - 320*q^5
- 528*q^6 - 861*q^7 - 1383*q^8 - 2218*q^9 - 3472*q^10 - 5451*q^11 -
8447*q^12 - 13020*q^13 - 20083*q^14 - 29512*q^15 - 39682*q^16 + O(q^17)

The following should give 0, but only approximately:

sage: q = X.parent().gen()
sage: E.defining_polynomial()(X,Y,1) + O(q^11) == 0
True

Note that below we have to change variable from x to q

sage: a1,_,a3,_,_=E.a_invariants()
sage: f=E.q_expansion(17)
sage: q=f.parent().gen()
sage: f/q == (X.derivative()/(2*Y+a1*X+a3))
True

modular_symbol( self, [sign=1], [normalize=True])

Return the modular symbol associated to this elliptic curve, with given sign and base ring. This is the map that sends r/s to a fixed multiple of 2*pi*I*f(z)dz from oo to r/s, normalized so that all values of this map take values in QQ.

If sign=1, the normalization is such that the p-adic L-function associated to this modular symbol is correct. I.e., the normalization is the same as for the integral period mapping divided by 2.

Input:

sign
- -1, or 1
base_ring
- a ring
normalize
- (default: True); if True, the modular symbol is correctly normalized (up to possibly a factor of -1 or 2). If False, the modular symbol is almost certainly not correctly normalized, i.e., all values will be a fixed scalar multiple of what they should be. But the initial computation of the modular symbol is much faster, though evaluation of it after computing it won't be any faster.

sage: E=EllipticCurve('37a1')
sage: M=E.modular_symbol(); M
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve
defined by y^2 + y = x^3 - x over Rational Field
sage: M(1/2)
0
sage: M(1/5)
1/2

modular_symbol_space( self, [sign=1], [base_ring=Rational Field], [bound=None])

Return the space of cuspidal modular symbols associated to this elliptic curve, with given sign and base ring.

Input:

sign
- 0, -1, or 1
base_ring
- a ring

sage: f = EllipticCurve('37b')
sage: f.modular_symbol_space()
Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 3 for Gamma_0(37) of weight 2 with sign 1 over Rational Field
sage: f.modular_symbol_space(-1)
Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 2 for Gamma_0(37) of weight 2 with sign -1 over Rational Field
sage: f.modular_symbol_space(0, bound=3)
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field

NOTE: If you just want the $ q$ -expansion, use self.q_expansion(prec).

mwrank( self, [options=])

Run Cremona's mwrank program on this elliptic curve and return the result as a string.

Input:

options
- string; passed when starting mwrank. The format is q p<precision> v<verbosity> b<hlim_q> x<naux> c<hlim_c> l t o s d>]

Output:
string
- output of mwrank on this curve

NOTE: The output is a raw string and completely illegible using automatic display, so it is recommended to use print for legible outout

sage: E = EllipticCurve('37a1')
sage: E.mwrank() #random
...
sage: print E.mwrank()
Curve [0,0,1,-1,0] :        Basic pair: I=48, J=-432
disc=255744
...
Generator 1 is [0:-1:1]; height 0.05111...

Regulator = 0.05111...

The rank and full Mordell-Weil basis have been determined unconditionally.
...

Options to mwrank can be passed:

sage: E = EllipticCurve([0,0,0,877,0])

Run mwrank with 'verbose' flag set to 0 but list generators if found

sage: print E.mwrank('-v0 -l')
Curve [0,0,0,877,0] :   0 <= rank <= 1
Regulator = 1

Run mwrank again, this time with a higher bound for point searching on homogeneous spaces:

sage: print E.mwrank('-v0 -l -b11')
Curve [0,0,0,877,0] :   Rank = 1
Generator 1 is [29604565304828237474403861024284371796799791624792913256602
210:-256256267988926809388776834045513089648669153204356603464786949:490078
023219787588959802933995928925096061616470779979261000]; height
95.980371987964
Regulator = 95.980371987964

mwrank_curve( self, [verbose=False])

Construct an mwrank_EllipticCurve from this elliptic curve

The resulting mwrank_EllipticCurve has available methods from John Cremona's eclib library.

sage: E=EllipticCurve('11a1')
sage: EE=E.mwrank_curve()
sage: EE
y^2+ y = x^3 - x^2 - 10*x - 20
sage: type(EE)
<class 'sage.libs.mwrank.interface.mwrank_EllipticCurve'>
sage: EE.isogeny_class()
([[0, -1, 1, -10, -20], [0, -1, 1, -7820, -263580], [0, -1, 1, 0, 0]],
[[0, 5, 5], [5, 0, 0], [5, 0, 0]])

newform( self)

Same as self.modular_form().

sage: E=EllipticCurve('37a1')
sage: E.newform()
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)
sage: E.newform() == E.modular_form()
True

ngens( self, [proof=None])

Return the number of generators of this elliptic curve.

NOTE: See gens() for further documentation. The function ngens() calls gens() if not already done, but only with default parameters. Better results may be obtained by calling mwrank() with carefully chosen parameters.

sage: E=EllipticCurve('37a1')
sage: E.ngens()
1

TO DO: This example should not cause a run-time error.

sage: E=EllipticCurve([0,0,0,877,0])
sage: # E.ngens()  ######## causes run-time error

sage: print E.mwrank('-v0 -b12 -l')
Curve [0,0,0,877,0] :   Rank = 1
Generator 1 is [29604565304828237474403861024284371796799791624792913256602
210:-256256267988926809388776834045513089648669153204356603464786949:490078
023219787588959802933995928925096061616470779979261000]; height
95.980371987964
Regulator = 95.980...

non_surjective( self, [A=1000])

Returns a list of primes p such that the mod-p representation $ \rho_{E,p}$ *might* not be surjective (this list usually contains 2, because of shortcomings of the algorithm). If p is not in the returned list, then rho_E,p is provably surjective (see A. Cojocaru's paper). If the curve has CM then infinitely many representations are not surjective, so we simply return the sequence [(0,"cm")] and do no further computation.

Input:

A
- an integer
Output:
list
- if curve has CM, returns [(0,"cm")]. Otherwise, returns a list of primes where mod-p representation very likely not surjective. At any prime not in this list, the representation is definitely surjective.

sage: E = EllipticCurve([0, 0, 1, -38, 90])  # 361A
sage: E.non_surjective()   # CM curve
[(0, 'cm')]

sage: E = EllipticCurve([0, -1, 1, 0, 0]) # X_1(11)
sage: E.non_surjective()
[(5, '5-torsion')]

sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A
sage: E.non_surjective()
[]

sage: E = EllipticCurve([0,-1,1,-2,-1])   # 141C
sage: E.non_surjective()
[(13, [1])]

ALGORITHM: When p<=3 use division polynomials. For 5 <= p <= B, where B is Cojocaru's bound, use the results in Section 2 of Serre's inventiones paper"Sur Les Representations Modulaires Deg Degre 2 de Galqbar Over Q."

Np( self, p)

The number of points on E modulo p, where p is a prime, not necessarily of good reduction. (When p is a bad prime, also counts the singular point.)

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.Np(2)
5
sage: E.Np(3)
5
sage: E.conductor()
11
sage: E.Np(11)
11

ordinary_primes( self, B)

Return a list of all ordinary primes for this elliptic curve up to and possibly including B.

sage: e = EllipticCurve('11a')
sage: e.aplist(20)
[-2, -1, 1, -2, 1, 4, -2, 0]
sage: e.ordinary_primes(97)
[3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97]
sage: e = EllipticCurve('49a')
sage: e.aplist(20)
[1, 0, 0, 0, 4, 0, 0, 0]
sage: e.supersingular_primes(97)
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97]
sage: e.ordinary_primes(97)
[2, 11, 23, 29, 37, 43, 53, 67, 71, 79]
sage: e.ordinary_primes(3)
[2]
sage: e.ordinary_primes(2)
[2]
sage: e.ordinary_primes(1)
[]

p_isogenous_curves( self, [p=None])

Return a list of pairs $ (p, L)$ where $ p$ is a prime and $ L$ is a list of the elliptic curves over $ \mathbf{Q}$ that are $ p$ -isogenous to this elliptic curve.

Input:

p
- prime or None (default: None); if a prime, returns a list of the p-isogenous curves. Otherwise, returns a list of all prime-degree isogenous curves sorted by isogeny degree.

This is implemented using Cremona's GP script allisog.gp.

sage: E = EllipticCurve([0,-1,0,-24649,1355209])   
sage: E.p_isogenous_curves()
[(2, [Elliptic Curve defined by y^2  = x^3 - x^2 - 91809*x - 9215775 over
Rational Field, Elliptic Curve defined by y^2  = x^3 - x^2 - 383809*x +
91648033 over Rational Field, Elliptic Curve defined by y^2  = x^3 - x^2 +
1996*x + 102894 over Rational Field])]

The isogeny class of the curve 11a2 has three curves in it. But p_isogenous_curves only returns one curves, since there is only one curve $ 5$ -isogenous to 11a2.

sage: E = EllipticCurve('11a2')
sage: E.p_isogenous_curves()
[(5, [Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over
Rational Field])]
sage: E.p_isogenous_curves(5)
[Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational
Field]
sage: E.p_isogenous_curves(3)
[]

In contrast, the curve 11a1 admits two $ 5$ -isogenies:

sage: E = EllipticCurve('11a1')
sage: E.p_isogenous_curves(5)
[Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over
Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 - x^2 over Rational Field]

padic_E2( self, p, [prec=20], [check=False], [check_hypotheses=True], [algorithm=auto])

Returns the value of the $ p$ -adic modular form $ E2$ for $ (E, \omega)$ where $ \omega$ is the usual invariant differential $ dx/(2y + a_1 x + a_3)$ .

Input:

p
- prime (>= 5) for which $ E$ is good and ordinary
prec
- (relative) p-adic precision (>= 1) for result
check
- boolean, whether to perform a consistency check. This will slow down the computation by a constant factor < 2. (The consistency check is to compute the whole matrix of frobenius on Monsky-Washnitzer cohomology, and verify that its trace is correct to the specified precision. Otherwise, the trace is used to compute one column from the other one (possibly after a change of basis).)
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic sigma function makes sense
algorithm
- one of "standard", "sqrtp", or "auto". This selects which version of Kedlaya's algorithm is used. The "standard" one is the one described in Kedlaya's paper. The "sqrtp" one has better performance for large $ p$ , but only works when $ p > 6N$ ($ N=$ prec). The "auto" option selects "sqrtp" whenever possible.

Note that if the "sqrtp" algorithm is used, a consistency check will automatically be applied, regardless of the setting of the "check" flag.

Output: p-adic number to precision prec

NOTES: - If the discriminant of the curve has nonzero valuation at p, then the result will not be returned mod $ p^$prec , but it still *will* have prec *digits* of precision.

TODO: - Once we have a better implementation of the "standard" algorithm, the algorithm selection strategy for "auto" needs to be revisited.

Author: David Harvey (2006-09-01): partly based on code written by Robert Bradshaw at the MSRI 2006 modular forms workshop

ACKNOWLEDGMENT: - discussion with Eyal Goren that led to the trace trick.

Here is the example discussed in the paper ``Computation of p-adic Heights and Log Convergence'' (Mazur, Stein, Tate):

sage: EllipticCurve([-1, 1/4]).padic_E2(5)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 +
2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + O(5^20)

Let's try to higher precision (this is the same answer the MAGMA implementation gives):

sage: EllipticCurve([-1, 1/4]).padic_E2(5, 100)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 +
2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + 4*5^20 +
5^21 + 4*5^22 + 2*5^23 + 3*5^24 + 3*5^26 + 2*5^27 + 3*5^28 + 2*5^30 + 5^31
+ 4*5^33 + 3*5^34 + 4*5^35 + 5^36 + 4*5^37 + 4*5^38 + 3*5^39 + 4*5^41 +
2*5^42 + 3*5^43 + 2*5^44 + 2*5^48 + 3*5^49 + 4*5^50 + 2*5^51 + 5^52 +
4*5^53 + 4*5^54 + 3*5^55 + 2*5^56 + 3*5^57 + 4*5^58 + 4*5^59 + 5^60 +
3*5^61 + 5^62 + 4*5^63 + 5^65 + 3*5^66 + 2*5^67 + 5^69 + 2*5^70 + 3*5^71 +
3*5^72 + 5^74 + 5^75 + 5^76 + 3*5^77 + 4*5^78 + 4*5^79 + 2*5^80 + 3*5^81 +
5^82 + 5^83 + 4*5^84 + 3*5^85 + 2*5^86 + 3*5^87 + 5^88 + 2*5^89 + 4*5^90 +
4*5^92 + 3*5^93 + 4*5^94 + 3*5^95 + 2*5^96 + 4*5^97 + 4*5^98 + 2*5^99 +
O(5^100)

Check it works at low precision too:

sage: EllipticCurve([-1, 1/4]).padic_E2(5, 1)
2 + O(5)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 2)
2 + 4*5 + O(5^2)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 3)
2 + 4*5 + O(5^3)

TODO: With the old(-er), i.e., <= sage-2.4 p-adics we got $ 5 + O(5^2)$ as output, i.e., relative precision 1, but with the newer p-adics we get relative precision 0 and absolute precision 1.

sage: EllipticCurve([1, 1, 1, 1, 1]).padic_E2(5, 1)
O(5)

Check it works for different models of the same curve (37a), even when the discriminant changes by a power of p (note that E2 depends on the differential too, which is why it gets scaled in some of the examples below):

sage: X1 = EllipticCurve([-1, 1/4])
sage: X1.j_invariant(), X1.discriminant()
 (110592/37, 37)
sage: X1.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

sage: X2 = EllipticCurve([0, 0, 1, -1, 0])
sage: X2.j_invariant(), X2.discriminant()
 (110592/37, 37)
sage: X2.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

sage: X3 = EllipticCurve([-1*(2**4), 1/4*(2**6)])
sage: X3.j_invariant(), X3.discriminant() / 2**12
 (110592/37, 37)
sage: 2**(-2) * X3.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

sage: X4 = EllipticCurve([-1*(7**4), 1/4*(7**6)])
sage: X4.j_invariant(), X4.discriminant() / 7**12
 (110592/37, 37)
sage: 7**(-2) * X4.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

sage: X5 = EllipticCurve([-1*(5**4), 1/4*(5**6)])
sage: X5.j_invariant(), X5.discriminant() / 5**12
 (110592/37, 37)
sage: 5**(-2) * X5.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

sage: X6 = EllipticCurve([-1/(5**4), 1/4/(5**6)])
sage: X6.j_invariant(), X6.discriminant() * 5**12
 (110592/37, 37)
sage: 5**2 * X6.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

Test check=True vs check=False:

sage: EllipticCurve([-1, 1/4]).padic_E2(5, 1, check=False)
2 + O(5)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 1, check=True)
2 + O(5)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 30, check=False)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 +
2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + 4*5^20 +
5^21 + 4*5^22 + 2*5^23 + 3*5^24 + 3*5^26 + 2*5^27 + 3*5^28 + O(5^30)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 30, check=True)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 +
2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + 4*5^20 +
5^21 + 4*5^22 + 2*5^23 + 3*5^24 + 3*5^26 + 2*5^27 + 3*5^28 + O(5^30)

Here's one using the $ p^{1/2}$ algorithm:

sage: EllipticCurve([-1, 1/4]).padic_E2(3001, 3, algorithm="sqrtp")
1907 + 2819*3001 + 1124*3001^2 + O(3001^3)

padic_height( self, p, [prec=20], [sigma=None], [check_hypotheses=True])

Computes the cyclotomic p-adic height.

The equation of the curve must be minimal at $ p$ .

Input:

p
- prime >= 5 for which the curve has semi-stable reduction
prec
- integer >= 1, desired precision of result
sigma
- precomputed value of sigma. If not supplied, this function will call padic_sigma to compute it.
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic height makes sense

Output: A function that accepts two parameters: * a Q-rational point on the curve whose height should be computed * optional boolean flag "check": if False, it skips some input checking, and returns the p-adic height of that point to the desired precision.

Author Log:

sage: E = EllipticCurve("37a")
sage: P = E.gens()[0]
sage: h = E.padic_height(5, 10)
sage: h(P)
4*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + 4*5^8 + 3*5^9 + O(5^10)

An anomalous case:

sage: h = E.padic_height(53, 10)
sage: h(P)
27*53^-1 + 22 + 32*53 + 5*53^2 + 42*53^3 + 20*53^4 + 43*53^5 + 30*53^6 +
17*53^7 + 22*53^8 + 35*53^9 + O(53^10)

Boundary case:

sage: E.padic_height(5, 3)(P)
4*5 + 3*5^2 + O(5^3)

A case that works the division polynomial code a little harder:

sage: E.padic_height(5, 10)(5*P)
4*5^3 + 3*5^4 + 3*5^5 + 4*5^6 + 4*5^7 + 5^8 + O(5^10)

Check that answers agree over a range of precisions:

sage: max_prec = 30    # make sure we get past p^2    # long time
sage: full = E.padic_height(5, max_prec)(P)           # long time
sage: for prec in range(1, max_prec):                 # long time
...       assert E.padic_height(5, prec)(P) == full   # long time

A supersingular prime for a curve:

sage: E = EllipticCurve('37a')
sage: E.is_supersingular(3)
True
sage: h = E.padic_height(3, 5)
sage: h(E.gens()[0])
(2*3 + 2*3^2 + 3^3 + 2*3^4 + 2*3^5 + O(3^6), 3^2 + 3^3 + 3^4 + 3^5 +
O(3^7))
sage: E.padic_regulator(5)                          
4*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + 4*5^8 + 3*5^9 + 3*5^10 + 5^11 +
5^12 + 3*5^13 + 3*5^15 + 2*5^16 + 3*5^17 + 2*5^18 + O(5^20)
sage: E.padic_regulator(3, 5)
(2*3 + O(3^3), 2*3^2 + O(3^4))

A torsion point in both the good and supersingular cases:

sage: E = EllipticCurve('11a')
sage: P = E.torsion_subgroup().gens()[0]; P
(5 : 5 : 1)
sage: h = E.padic_height(19, 5)
sage: h(P)
0
sage: h = E.padic_height(5, 5)
sage: h(P)
0

padic_height_pairing_matrix( self, p, [prec=20], [height=None], [check_hypotheses=True])

Computes the cyclotomic $ p$ -adic height pairing matrix of this curve with respect to the basis self.gens() for the Mordell-Weil group for a given odd prime p of good ordinary reduction.

This curve must be in minimal weierstrass form.

Input:

p
- prime >= 5
prec
- answer will be returned modulo $ p^{\var{prec}}$
height
- precomputed height function. If not supplied, this function will call padic_height to compute it.
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic height makes sense

Output: The p-adic cyclotomic height pairing matrix of this curve to the given precision.

TODO: - remove restriction that curve must be in minimal weierstrass form. This is currently required for E.gens().

Author Log:

sage: E = EllipticCurve("37a")
sage: E.padic_height_pairing_matrix(5, 10)
[4*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + 4*5^8 + 3*5^9 + O(5^10)]

A rank two example:

sage: e =EllipticCurve('389a')
sage: e._set_gens([e(-1, 1), e(1,0)])  # avoid platform dependent gens
sage: e.padic_height_pairing_matrix(5,10)
[2*5 + 2*5^2 + 4*5^3 + 3*5^4 + 3*5^5 + 4*5^6 + 3*5^7 + 4*5^8 + O(5^10)     
4*5 + 3*5^3 + 2*5^4 + 5^5 + 3*5^7 + 3*5^8 + 2*5^9 + O(5^10)]
[          4*5 + 3*5^3 + 2*5^4 + 5^5 + 3*5^7 + 3*5^8 + 2*5^9 + O(5^10)     
5 + 4*5^2 + 4*5^3 + 2*5^4 + 4*5^5 + 5^6 + 4*5^9 + O(5^10)]

An anomalous rank 3 example:

sage: e = EllipticCurve("5077a")
sage: e._set_gens([e(-1,3), e(2,0), e(4,6)])
sage: e.padic_height_pairing_matrix(5,4)
[                1 + 5 + O(5^4)       1 + 4*5 + 2*5^3 + O(5^4)          
2*5 + 3*5^3 + O(5^4)]
[      1 + 4*5 + 2*5^3 + O(5^4)       2 + 5^2 + 3*5^3 + O(5^4)     3 +
4*5^2 + 4*5^3 + O(5^4)]
[          2*5 + 3*5^3 + O(5^4)     3 + 4*5^2 + 4*5^3 + O(5^4) 4 + 5 +
3*5^2 + 3*5^3 + O(5^4)]

padic_height_via_multiply( self, p, [prec=20], [E2=None], [check_hypotheses=True])

Computes the cyclotomic p-adic height.

The equation of the curve must be minimal at $ p$ .

Input:

p
- prime >= 5 for which the curve has good ordinary reduction
prec
- integer >= 2, desired precision of result
E2
- precomputed value of E2. If not supplied, this function will call padic_E2 to compute it. The value supplied must be correct mod $ p^(prec-2)$ (or slightly higher in the anomalous case; see the code for details).
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic height makes sense

Output: A function that accepts two parameters: * a Q-rational point on the curve whose height should be computed * optional boolean flag "check": if False, it skips some input checking, and returns the p-adic height of that point to the desired precision.

Author: David Harvey (2008-01): based on the padic_height() function, using the algorithm of ``Computing p-adic heights via point multiplication''

sage: E = EllipticCurve("37a")
sage: P = E.gens()[0]
sage: h = E.padic_height_via_multiply(5, 10)
sage: h(P)
4*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + 4*5^8 + 3*5^9 + O(5^10)

An anomalous case:

sage: h = E.padic_height_via_multiply(53, 10)
sage: h(P)
27*53^-1 + 22 + 32*53 + 5*53^2 + 42*53^3 + 20*53^4 + 43*53^5 + 30*53^6 +
17*53^7 + 22*53^8 + 35*53^9 + O(53^10)

Supply the value of E2 manually:

sage: E2 = E.padic_E2(5, 8)
sage: E2
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + O(5^8)
sage: h = E.padic_height_via_multiply(5, 10, E2=E2)
sage: h(P)
4*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + 4*5^8 + 3*5^9 + O(5^10)

Boundary case:

sage: E.padic_height_via_multiply(5, 3)(P)
4*5 + 3*5^2 + O(5^3)

Check that answers agree over a range of precisions:

sage: max_prec = 30    # make sure we get past p^2    # long time
sage: full = E.padic_height(5, max_prec)(P)           # long time
sage: for prec in range(2, max_prec):                 # long time
...       assert E.padic_height_via_multiply(5, prec)(P) == full   # long
time

padic_lseries( self, p, [normalize=True], [use_eclib=False])

Return the $ p$ -adic $ L$ -series of self at $ p$ , which is an object whose approx method computes approximation to the true $ p$ -adic $ L$ -series to any deesired precision.

Input:

p
- prime
normalize
- (default: True); if True the p-adic L-series is normalized correctly (up to multiplication by -1 and 2); otherwise it isn't, but computation of the series is quicker.

sage: E = EllipticCurve('37a')
sage: L = E.padic_lseries(5); L
5-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over
Rational Field
sage: type(L)
<class 'sage.schemes.elliptic_curves.padic_lseries.pAdicLseriesOrdinary'>

We compute the $ 3$ -adic $ L$ -series of two curves of rank 0 and in each case verify the interpolation property for their leading coefficient (i.e., value at 0):

sage: e = EllipticCurve('11a')
sage: ms = e.modular_symbol()
sage: [ms(1/11), ms(1/3), ms(0), ms(oo)]
[0, -3/10, 1/5, 0]
sage: ms(0)
1/5
sage: L = e.padic_lseries(3)
sage: P = L.series(5)
sage: P(0)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7)
sage: alpha = L.alpha(9); alpha
2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + O(3^9)
sage: R.<x> = QQ[]
sage: f = x^2 - e.ap(3)*x + 3
sage: f(alpha)
O(3^9)
sage: r = e.lseries().L_ratio(); r
1/5
sage: (1 - alpha^(-1))^2 * r
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + O(3^9)
sage: P(0)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7)

Next consider the curve 37b:

sage: e = EllipticCurve('37b')
sage: L = e.padic_lseries(3)
sage: P = L.series(5)
sage: alpha = L.alpha(9); alpha
1 + 2*3 + 3^2 + 2*3^5 + 2*3^7 + 3^8 + O(3^9)
sage: r = e.lseries().L_ratio(); r
1/3
sage: (1 - alpha^(-1))^2 * r
3 + 3^2 + 2*3^4 + 2*3^5 + 2*3^6 + 3^7 + O(3^9)
sage: P(0)
3 + 3^2 + 2*3^4 + 2*3^5 + O(3^6)

We can use eclib to compute the $ L$ -series. (but we don't normalize the result yet)

sage: e = EllipticCurve('11a')
sage: L = e.padic_lseries(3,normalize=False,use_eclib=True)
sage: P1 = L.series(5,10)
sage: L = e.padic_lseries(3,normalize=False,use_eclib=False)
sage: P2 = L.series(5,10)
sage: P2*= P1(0) / P2(0)  #rescale P2 such that the constant term agrees with P1
sage: T = P1.parent().gen()
sage: Q = sum([O(3^5)*T^i for i in range(9)]) + O(T^9) #essentially zero
sage: Q == Q + (P1 - P2)                   #check that every term agrees up to O(3^4)
True

padic_regulator( self, p, [prec=20], [height=None], [check_hypotheses=True])

Computes the cyclotomic p-adic regulator of this curve.

This curve must be in minimal weierstrass form.

Input:

p
- prime >= 5
prec
- answer will be returned modulo $ p^{\var{prec}}$
height
- precomputed height function. If not supplied, this function will call padic_height to compute it.
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic height makes sense

Output: The p-adic cyclotomic regulator of this curve, to the requested precision.

If the rank is 0, we output 1.

TODO: - remove restriction that curve must be in minimal weierstrass form. This is currently required for E.gens().

Author Log:

sage: E = EllipticCurve("37a")
sage: E.padic_regulator(5, 10)
4*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + 4*5^8 + 3*5^9 + O(5^10)

An anomalous case:

sage: E.padic_regulator(53, 10)
27*53^-1 + 22 + 32*53 + 5*53^2 + 42*53^3 + 20*53^4 + 43*53^5 + 30*53^6 +
17*53^7 + 22*53^8 + O(53^9)

An anomalous case where the precision drops some:

sage: E = EllipticCurve("5077a")
sage: E.padic_regulator(5, 10)
4*5 + 3*5^2 + 2*5^4 + 2*5^5 + 2*5^6 + 2*5^8 + 3*5^9 + O(5^10)

Check that answers agree over a range of precisions:

sage: max_prec = 30    # make sure we get past p^2    # long time
sage: full = E.padic_regulator(5, max_prec)           # long time
sage: for prec in range(1, max_prec):                 # long time
...       assert E.padic_regulator(5, prec) == full   # long time

padic_sigma( self, p, [N=20], [E2=None], [check=False], [check_hypotheses=True])

Computes the p-adic sigma function with respect to the standard invariant differential $ dx/(2y + a_1 x + a_3)$ , as defined by Mazur and Tate, as a power series in the usual uniformiser $ t$ at the origin.

The equation of the curve must be minimal at $ p$ .

Input:

p
- prime >= 5 for which the curve has good ordinary reduction
N
- integer >= 1, indicates precision of result; see OUTPUT section for description
E2
- precomputed value of E2. If not supplied, this function will call padic_E2 to compute it. The value supplied must be correct mod $ p^{N-2}$ .
check
- boolean, whether to perform a consistency check (i.e. verify that the computed sigma satisfies the defining
differential equation
- note that this does NOT guarantee correctness of all the returned digits, but it comes pretty close :-))
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic sigma function makes sense

Output: A power series $ t + \cdots$ with coefficients in $ \mathbf{Z}_p$ .

The output series will be truncated at $ O(t^{N+1})$ , and the coefficient of $ t^n$ for $ n \geq 1$ will be correct to precision $ O(p^{N-n+1})$ .

In practice this means the following. If $ t_0 = p^k u$ , where $ u$ is a $ p$ -adic unit with at least $ N$ digits of precision, and $ k \geq 1$ , then the returned series may be used to compute $ \sigma(t_0)$ correctly modulo $ p^{N+k}$ (i.e. with $ N$ correct $ p$ -adic digits).

ALGORITHM: Described in ``Efficient Computation of p-adic Heights'' (David Harvey), which is basically an optimised version of the algorithm from ``p-adic Heights and Log Convergence'' (Mazur, Stein, Tate).

Running time is soft- $ O(N^2 \log p)$ , plus whatever time is necessary to compute $ E_2$ .

Author Log:

sage: EllipticCurve([-1, 1/4]).padic_sigma(5, 10)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7
+ O(5^8))*t^3 + O(5^7)*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6))*t^5 +
O(5^5)*t^6 + (2 + 2*5 + 5^2 + 4*5^3 + O(5^4))*t^7 + O(5^3)*t^8 + (1 + 2*5 +
O(5^2))*t^9 + O(5)*t^10 + O(t^11)

Run it with a consistency check:

sage: EllipticCurve("37a").padic_sigma(5, 10, check=True)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7
+ O(5^8))*t^3 + (3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 +
O(5^7))*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6))*t^5 + (2 + 3*5 + 5^4
+ O(5^5))*t^6 + (4 + 3*5 + 2*5^2 + O(5^4))*t^7 + (2 + 3*5 + 2*5^2 +
O(5^3))*t^8 + (4*5 + O(5^2))*t^9 + (1 + O(5))*t^10 + O(t^11)

Boundary cases:

sage: EllipticCurve([1, 1, 1, 1, 1]).padic_sigma(5, 1)
 (1 + O(5))*t + O(t^2)
sage: EllipticCurve([1, 1, 1, 1, 1]).padic_sigma(5, 2)
 (1 + O(5^2))*t + (3 + O(5))*t^2 + O(t^3)

Supply your very own value of E2:

sage: X = EllipticCurve("37a")
sage: my_E2 = X.padic_E2(5, 8)
sage: my_E2 = my_E2 + 5**5    # oops!!!
sage: X.padic_sigma(5, 10, E2=my_E2)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 4*5^5 + 2*5^6
+ 3*5^7 + O(5^8))*t^3 + (3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 +
O(5^7))*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 3*5^5 + O(5^6))*t^5 + (2 + 3*5 +
5^4 + O(5^5))*t^6 + (4 + 3*5 + 2*5^2 + O(5^4))*t^7 + (2 + 3*5 + 2*5^2 +
O(5^3))*t^8 + (4*5 + O(5^2))*t^9 + (1 + O(5))*t^10 + O(t^11)

Check that sigma is ``weight 1''.

sage: f = EllipticCurve([-1, 3]).padic_sigma(5, 10)
sage: g = EllipticCurve([-1*(2**4), 3*(2**6)]).padic_sigma(5, 10)
sage: t = f.parent().gen()
sage: f(2*t)/2
(1 + O(5^10))*t + (4 + 3*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 3*5^6 + 5^7 +
O(5^8))*t^3 + (3 + 3*5^2 + 5^4 + 2*5^5 + O(5^6))*t^5 + (4 + 5 + 3*5^3 +
O(5^4))*t^7 + (4 + 2*5 + O(5^2))*t^9 + O(5)*t^10 + O(t^11)
sage: g
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (4 + 3*5 + 3*5^2 + 3*5^3 + 4*5^4 +
4*5^5 + 3*5^6 + 5^7 + O(5^8))*t^3 + O(5^7)*t^4 + (3 + 3*5^2 + 5^4 + 2*5^5 +
O(5^6))*t^5 + O(5^5)*t^6 + (4 + 5 + 3*5^3 + O(5^4))*t^7 + O(5^3)*t^8 + (4 +
2*5 + O(5^2))*t^9 + O(5)*t^10 + O(t^11)
sage: f(2*t)/2 -g
O(t^11)

Test that it returns consistent results over a range of precision:

sage: max_N = 30   # get up to at least p^2         # long time
sage: E = EllipticCurve([1, 1, 1, 1, 1])            # long time
sage: p = 5                                         # long time
sage: E2 = E.padic_E2(5, max_N)                     # long time
sage: max_sigma = E.padic_sigma(p, max_N, E2=E2)    # long time
sage: for N in range(3, max_N):                     # long time
...      sigma = E.padic_sigma(p, N, E2=E2)         # long time
...      assert sigma == max_sigma

padic_sigma_truncated( self, p, [N=20], [lamb=0], [E2=None], [check_hypotheses=True])

Computes the p-adic sigma function with respect to the standard invariant differential $ dx/(2y + a_1 x + a_3)$ , as defined by Mazur and Tate, as a power series in the usual uniformiser $ t$ at the origin.

The equation of the curve must be minimal at $ p$ .

This function differs from padic_sigma() in the precision profile of the returned power series; see OUTPUT below.

Input:

p
- prime >= 5 for which the curve has good ordinary reduction
N
- integer >= 2, indicates precision of result; see OUTPUT section for description
lamb
- integer >= 0, see OUTPUT section for description
E2
- precomputed value of E2. If not supplied, this function will call padic_E2 to compute it. The value supplied must be correct mod $ p^{N-2}$ .
check_hypotheses
- boolean, whether to check that this is a curve for which the p-adic sigma function makes sense

Output: A power series $ t + \cdots$ with coefficients in $ \mathbf{Z}_p$ .

The coefficient of $ t^j$ for $ j \geq 1$ will be correct to precision $ O(p^{N - 2 + (3 - j)(lamb + 1)})$ .

ALGORITHM: Described in ``Efficient Computation of p-adic Heights'' (David Harvey, to appear in LMS JCM), which is basically an optimised version of the algorithm from ``p-adic Heights and Log Convergence'' (Mazur, Stein, Tate), and ``Computing p-adic heights via point multiplication'' (David Harvey, still draft form).

Running time is soft- $ O(N^2 \lambda^{-1} \log p)$ , plus whatever time is necessary to compute $ E_2$ .

Author: David Harvey (2008-01): wrote based on previous padic_sigma function

sage: E = EllipticCurve([-1, 1/4])
sage: E.padic_sigma_truncated(5, 10)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7
+ O(5^8))*t^3 + O(5^7)*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6))*t^5 +
O(5^5)*t^6 + (2 + 2*5 + 5^2 + 4*5^3 + O(5^4))*t^7 + O(5^3)*t^8 + (1 + 2*5 +
O(5^2))*t^9 + O(5)*t^10 + O(t^11)

Note the precision of the $ t^3$ coefficient depends only on $ N$ , not on lamb:

sage: E.padic_sigma_truncated(5, 10, lamb=2)
O(5^17) + (1 + O(5^14))*t + O(5^11)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 +
4*5^7 + O(5^8))*t^3 + O(5^5)*t^4 + (2 + O(5^2))*t^5 + O(t^6)

Compare against plain padic_sigma() function over a dense range of N and lamb

sage: E = EllipticCurve([1, 2, 3, 4, 7])                            # long time
sage: E2 = E.padic_E2(5, 50)                                        # long time
sage: for N in range(2, 10):                                        # long time
...      for lamb in range(10):                                     # long
time
...         correct = E.padic_sigma(5, N + 3*lamb, E2=E2)           # long
time
...         compare = E.padic_sigma_truncated(5, N=N, lamb=lamb, E2=E2)   
# long time
...         assert compare == correct                               # long
time

pari_curve( self, [prec=None], [factor=1])

Return the PARI curve corresponding to this elliptic curve.

Input:

prec
- The precision of quantities calculated for the returned curve (in decimal digits). if None, defaults to factor * the precision of the largest cached curve (or 10 if none yet computed)
factor
- the factor to increase the precision over the maximum previously computed precision. Only used if prec (which gives an explicit precision) is None.

sage: E = EllipticCurve([0, 0,1,-1,0])
sage: e = E.pari_curve()
sage: type(e)
<type 'sage.libs.pari.gen.gen'>
sage: e.type()
't_VEC'
sage: e.ellan(10)
[1, -2, -3, 2, -2, 6, -1, 0, 6, 4]

sage: E = EllipticCurve(RationalField(), ['1/3', '2/3'])
sage: e = E.pari_curve()
sage: e.type()
't_VEC'
sage: e[:5]
[0, 0, 0, 1/3, 2/3]

pari_mincurve( self, [prec=None])

Return the PARI curve corresponding to a minimal model for this elliptic curve.

Input:

prec
- The precision of quantities calculated for the returned curve (in decimal digits). if None, defaults to the precision of the largest cached curve (or 10 if none yet computed)

sage: E = EllipticCurve(RationalField(), ['1/3', '2/3'])
sage: e = E.pari_mincurve()
sage: e[:5]
[0, 0, 0, 27, 486]
sage: E.conductor()
47232
sage: e.ellglobalred()
[47232, [1, 0, 0, 0], 2]

period_lattice( self)

Returns the period lattice of the elliptic curve.

sage: E = EllipticCurve('37a')
sage: E.period_lattice()
Period lattice associated to Elliptic Curve defined by y^2 + y = x^3 - x
over Rational Field

point_search( self, height_limit, [verbose=True])

Search for points on a curve up to an input bound on the naive logarithmic height.

Input:

height_limit (float)
- bound on naive height (at most 21,
or mwrank overflows
- see below)

verbose (bool)
- (default: True)

If True, report on each point as found together with linear relations between the points found and the saturation process.

If False, just return the result.

Output: points (list) - list of independent points which generate the subgroup of the Mordell-Weil group generated by the points found and then p-saturated for p<20.

WARNING: height_limit is logarithmic, so increasing by 1 will cause the running time to increase by a factor of approximately 4.5 (=exp(1.5)). The limit of 21 is to prevent overflow, but in any case using height_limit=20 takes rather a long time!

IMPLEMENTATION: Uses Cremona's mwrank package. At the heart of this function is Cremona's port of Stoll's ratpoints program (version 1.4).

sage: E=EllipticCurve('389a1')
sage: E.point_search(5, verbose=False)
[(0 : -1 : 1), (-1 : 1 : 1)]

Increasing the height_limit takes longer, but finds no more points:

sage: E.point_search(10, verbose=False)
[(0 : -1 : 1), (-1 : 1 : 1)]

In fact this curve has rank 2 so no more than 2 points will ever be output, but we are not using this fact.

sage: E.saturation(_)
([(0 : -1 : 1), (-1 : 1 : 1)], '1', 0.152460172772408)

What this shows is that if the rank is 2 then the points listed do generate the Mordell-Weil group (mod torsion). Finally,

sage: E.rank()
2

q_eigenform( self, prec)

Synonym for self.q_expansion(prec).

sage: E=EllipticCurve('37a1')
sage: E.q_eigenform(10)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + O(q^10)
sage: E.q_eigenform(10) == E.q_expansion(10)
True

q_expansion( self, prec)

Return the $ q$ -expansion to precision prec of the newform attached to this elliptic curve.

Input:

prec
- an integer

Output: a power series (in th evariable 'q')

NOTE: If you want the output to be a modular form and not just a $ q$ -expansion, use self.modular_form().

sage: E=EllipticCurve('37a1')
sage: E.q_expansion(20)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - 5*q^11 -
6*q^12 - 2*q^13 + 2*q^14 + 6*q^15 - 4*q^16 - 12*q^18 + O(q^20)

quadratic_twist( self, D)

Return the global minimal model of the quadratic twist of this curve by D.

sage: E=EllipticCurve('37a1')
sage: E7=E.quadratic_twist(7); E7
Elliptic Curve defined by y^2  = x^3 - 784*x + 5488 over Rational Field
sage: E7.conductor()
29008
sage: E7.quadratic_twist(7) == E
True

rank( self, [use_database=False], [verbose=False], [only_use_mwrank=True], [algorithm=mwrank_shell], [proof=None])

Return the rank of this elliptic curve, assuming no conjectures.

If we fail to provably compute the rank, raises a RuntimeError exception.

Input:

use_database (bool)
- (default: False), if True, try to look up the regulator in the Cremona database.
verbose
- (default: None), if specified changes the verbosity of mwrank computations.
algorithm
- 'mwrank_shell' - call mwrank shell command
- 'mwrank_lib' - call mwrank c library
only_use_mwrank
- (default: True) if False try using analytic rank methods first.
proof
- bool or None (default: None, see proof.elliptic_curve or sage.structure.proof). Note that results obtained from databases are considered proof = True

Output:
rank (int)
- the rank of the elliptic curve.

IMPLEMENTATION: Uses L-functions, mwrank, and databases.

sage: EllipticCurve('11a').rank()
0
sage: EllipticCurve('37a').rank()
1
sage: EllipticCurve('389a').rank()
2
sage: EllipticCurve('5077a').rank()
3
sage: EllipticCurve([1, -1, 0, -79, 289]).rank()   # long time.  This will use the default proof behavior of True.
4
sage: EllipticCurve([0, 0, 1, -79, 342]).rank(proof=False)  # long time -- but under a minute
5
sage: EllipticCurve([0, 0, 1, -79, 342]).simon_two_descent()[0]  # much faster -- almost instant. 
5

Examples with denominators in defining equations:

sage: E = EllipticCurve( [0, 0, 0, 0, -675/4])
sage: E.rank()
0
sage: E = EllipticCurve( [0, 0, 1/2, 0, -1/5])
sage: E.rank()
1
sage: E.minimal_model().rank()
1

real_components( self)

Returns 1 if there is 1 real component and 2 if there are 2.

sage: E = EllipticCurve('37a')
sage: E.real_components ()
2
sage: E = EllipticCurve('37b')
sage: E.real_components ()
2
sage: E = EllipticCurve('11a')
sage: E.real_components ()
1

reducible_primes( self)

Returns a list of the primes $ p$ such that the mod $ p$ representation $ \rho_{E,p}$ is reducible. For all other primes the representation is irreducible.

NOTE - this is not provably correct in general. See the documentation for self.isogeny_class.

sage: E = EllipticCurve('225a')
sage: E.reducible_primes()
[3]

regulator( self, [use_database=True], [verbose=None], [proof=None])

Returns the regulator of this curve, which must be defined over Q.

Input:

use_database
- bool (default: False), if True, try to look up the regulator in the Cremona database.
verbose
- (default: None), if specified changes the verbosity of mwrank computations.
proof
- bool or None (default: None, see proof.[tab] or sage.structure.proof). Note that results from databases are considered proof = True

sage: E = EllipticCurve([0, 0, 1, -1, 0])
sage: E.regulator()              # long time (1 second)
0.0511114082399688
sage: EllipticCurve('11a').regulator()
1.00000000000000
sage: EllipticCurve('37a').regulator()
0.0511114082399688
sage: EllipticCurve('389a').regulator()
0.152460177943144
sage: EllipticCurve('5077a').regulator()    # random low order bit
0.417143558758385
sage: EllipticCurve([1, -1, 0, -79, 289]).regulator()  # long time (seconds)
1.50434488827528
sage: EllipticCurve([0, 0, 1, -79, 342]).regulator(proof=False)  # long time (seconds)
14.7905275701310

root_number( self)

Returns the root number of this elliptic curve.

This is 1 if the order of vanishing of the L-function L(E,s) at 1 is even, and -1 if it is odd.

sage: EllipticCurve('11a1').root_number()
1
sage: EllipticCurve('37a1').root_number()
-1
sage: EllipticCurve('389a1').root_number()
1

satisfies_heegner_hypothesis( self, D)

Returns True precisely when D is a fundamental discriminant that satisfies the Heegner hypothesis for this elliptic curve.

sage: E = EllipticCurve('11a1')
sage: E.satisfies_heegner_hypothesis(-7)
True
sage: E.satisfies_heegner_hypothesis(-11)
False

saturation( self, points, [verbose=False], [max_prime=0], [odd_primes_only=False])

Given a list of rational points on E, compute the saturation in E(Q) of the subgroup they generate.

Input:

points (list)
- list of points on E
verbose (bool)
- (default: False), if True, give verbose output
max_prime (int)
- (default: 0), saturation is performed for all primes up to max_prime. If max_prime==0, perform saturation at *all* primes, i.e., compute the true saturation.
odd_primes_only (bool)
- only do saturation at odd primes

Output:
saturation (list)
- points that form a basis for the saturation
index (int)
- the index of the group generated by points in their saturation
regulator (float)
- regulator of saturated points.

IMPLEMENTATION: Uses Cremona's mwrank package. With max_prime=0, we call mwrank with successively larger prime bounds until the full saturation is provably found. The results of saturation at the previous primes is stored in each case, so this should be reasonably fast.

sage: E=EllipticCurve('37a1')
sage: P=E.gens()[0]
sage: Q=5*P; Q
(1/4 : -3/8 : 1)
sage: E.saturation([Q])
([(0 : -1 : 1)], '5', 0.0511114075779915)

sea( self, p, [early_abort=False])

Return the number of points on $ E$ over $ \mathbf{F}_p$ computed using the SEA algorithm, as implemented in PARI by Christophe Doche and Sylvain Duquesne.

Input:

p
- a prime number
early_abort
- bool (default: Falst); if True an early abort technique is used and the computation is interrupted as soon as a small divisor of the order is detected.

Note: As of 2006-02-02 this function does not work on Microsoft Windows under Cygwin (though it works under vmware of course).

sage: E = EllipticCurve('37a')
sage: E.sea(next_prime(10^30))
1000000000000001426441464441649

selmer_rank_bound( self)

Bound on the rank of the curve, computed using the 2-selmer group. This is the rank of the curve minus the rank of the 2-torsion, minus a number determined by whatever mwrank was able to determine related to Sha[2]. Thus in many cases, this is the actual rank of the curve.

The following is the curve 960D1, which has rank 0, but Sha of order 4.

sage: E = EllipticCurve([0, -1, 0, -900, -10098])
sage: E.selmer_rank_bound()
0

It gives 0 instead of 2, because it knows Sha is nontrivial. In contrast, for the curve 571A, also with rank 0 and Sha of order 4, we get a worse bound:

sage: E = EllipticCurve([0, -1, 1, -929, -10595])
sage: E.selmer_rank_bound()
2
sage: E.rank(only_use_mwrank=False)   # uses L-function
0

sha( self)

Return an object of class 'sage.schemes.elliptic_curves.sha.Sha' attached to this elliptic curve.

This can be used in functions related to bounding the order of Sha (The Tate-Shafarevich group of the curve).

sage: E=EllipticCurve('37a1')
sage: S=E.sha()
sage: S
<class 'sage.schemes.elliptic_curves.sha.Sha'>
sage: S.bound_kolyvagin()
([2], 1)

silverman_height_bound( self)
Return the Silverman height bound. This is a positive real (floating point) number B such that for all rational points $ P$ on the curve,

$\displaystyle h(P) \le \hat{h}(P) + B
$

where h(P) is the logarithmic height of $ P$ and $ \hat{h}(P)$ is the canonical height.

Note that the CPS_height_bound is often better (i.e. smaller) than the Silverman bound.

sage: E=EllipticCurve('37a1')
sage: E.silverman_height_bound()
4.8254007581809182
sage: E.CPS_height_bound()
0.16397076103046915

simon_two_descent( self, [verbose=0], [lim1=5], [lim3=50], [limtriv=10], [maxprob=20], [limbigprime=30])

Given a curve with no 2-torsion, computes (probably) the rank of the Mordell-Weil group, with certainty the rank of the 2-Selmer group, and a list of independent points on the curve.

Input:

verbose
- integer, 0,1,2,3; (default: 0), the verbosity level
lim1
- (default: 5) limite des points triviaux sur les quartiques
lim3
- (default: 50) limite des points sur les quartiques ELS
limtriv
- (default: 10) limite des points triviaux sur la courbe elliptique
maxprob
- (default: 20)
limbigprime
- (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don't any probabilistic tests.

Output:
integer
- "probably" the rank of self
integer
- the 2-rank of the Selmer group
list
- list of independent points on the curve.

IMPLEMENTATION: Uses Denis Simon's GP/PARI scripts from http://www.math.unicaen.fr/~simon/

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

We compute the ranks of the curves of lowest known conductor up to rank $ 8$ . Amazingly, each of these computations finishes almost instantly!

sage: E = EllipticCurve('11a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(0, 0, [])
sage: E = EllipticCurve('37a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(1, 1, [(0 : 0 : 1)])
sage: E = EllipticCurve('389a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)])
sage: E = EllipticCurve('5077a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(3, 3, [(1 : 0 : 1), (2 : -1 : 1), (0 : 2 : 1)])

In this example Simon's program does not find any points, though it does correctly compute the rank of the 2-Selmer group.

sage: E = EllipticCurve([1, -1, 0, -751055859, -7922219731979])     # long (0.6 seconds)
sage: set_random_seed(0)
sage: E.simon_two_descent ()
(1, 1, [])

The rest of these entries were taken from Tom Womack's page http://tom.womack.net/maths/conductors.htm

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: set_random_seed(0)
sage: E.simon_two_descent()
(4, 4, [(6 : -5 : 1), (4 : 3 : 1), (5 : -3 : 1), (8 : -15 : 1)])
sage: E = EllipticCurve([0, 0, 1, -79, 342])
sage: set_random_seed(0)
sage: E.simon_two_descent()
(5, 5, [(5 : 8 : 1), (10 : 23 : 1), (3 : 11 : 1), (4 : -10 : 1), (0 : 18 :
1)])
sage: E = EllipticCurve([1, 1, 0, -2582, 48720])
sage: set_random_seed(0)
sage: r, s, G = E.simon_two_descent(); r,s
(6, 6)
sage: E = EllipticCurve([0, 0, 0, -10012, 346900])
sage: set_random_seed(0)
sage: r, s, G = E.simon_two_descent(); r,s
(7, 7)
sage: E = EllipticCurve([0, 0, 1, -23737, 960366])    
sage: set_random_seed(0)
sage: r, s, G = E.simon_two_descent(); r,s
(8, 8)

supersingular_primes( self, B)

Return a list of all supersingular primes for this elliptic curve up to and possibly including B.

sage: e = EllipticCurve('11a')
sage: e.aplist(20)
[-2, -1, 1, -2, 1, 4, -2, 0]
sage: e.supersingular_primes(1000)
[2, 19, 29, 199, 569, 809]

sage: e = EllipticCurve('27a')
sage: e.aplist(20)
[0, 0, 0, -1, 0, 5, 0, -7]
sage: e.supersingular_primes(97)
[2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89]
sage: e.ordinary_primes(97)
[7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97]
sage: e.supersingular_primes(3)
[2]
sage: e.supersingular_primes(2)
[2]
sage: e.supersingular_primes(1)
[]

tamagawa_number( self, p)

The Tamagawa number of the elliptic curve at $ p$ .

sage: E = EllipticCurve('11a')
sage: E.tamagawa_number(11)
5
sage: E = EllipticCurve('37b')
sage: E.tamagawa_number(37)
3

tamagawa_numbers( self)

Return a list of all Tamagawa numbers for all prime divisors of the conductor (in order).

sage: e = EllipticCurve('30a1')
sage: e.tamagawa_numbers()
[2, 3, 1]
sage: vector(e.tamagawa_numbers())
(2, 3, 1)

tamagawa_product( self)

Returns the product of the Tamagawa numbers.

sage: E = EllipticCurve('54a')
sage: E.tamagawa_product ()
3

tate_curve( self, p)

Creates the Tate Curve over the $ p$ -adics associated to this elliptic curves.

This Tate curve a $ p$ -adic curve with split multiplicative reduction of the form $ y^2+xy=x^3+s_4 x+s_6$ which is isomorphic to the given curve over the algebraic closure of $ \mathbf{Q}_p$ . Its points over $ \mathbf{Q}_p$ are isomorphic to $ \mathbf{Q}_p^{\times}/q^{\mathbf{Z}}$ for a certain parameter $ q\in\mathbf{Z}_p$ .

Input:

p - a prime where the curve has multiplicative reduction.

sage: e = EllipticCurve('130a1')
sage: e.tate_curve(2)
2-adic Tate curve associated to the Elliptic Curve defined by y^2 + x*y + y
= x^3 - 33*x + 68 over Rational Field

The input curve must have multiplicative reduction at the prime.

sage: e.tate_curve(3)
Traceback (most recent call last):
...
ValueError: The elliptic curve must have multiplicative reduction at 3

We compute with $ p=5$ :

sage: T = e.tate_curve(5); T
5-adic Tate curve associated to the Elliptic Curve defined by y^2 + x*y + y
= x^3 - 33*x + 68 over Rational Field

We find the Tate parameter $ q$ :

sage: T.parameter(prec=5)
3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8)

We compute the $ L$ -invariant of the curve:

sage: T.L_invariant(prec=10)
5^3 + 4*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 3*5^8 + 5^9 + O(5^10)

three_selmer_rank( self, [bound=0], [method=2])

Return the 3-selmer rank of this elliptic curve, computed using Magma.

This is not implemented for all curves; a NotImplementedError exception is raised when this function is called on curves for which 3-descent isn't implemented.

Note: Use a slightly modified version of Michael Stoll's MAGMA file 3descent.m. You must have Magma to use this function.

sage: EllipticCurve('37a').three_selmer_rank()  # optional \& long -- Magma
1

sage: EllipticCurve('14a1').three_selmer_rank()      # optional
Traceback (most recent call last):
...
NotImplementedError:  Currently, only the case with irreducible phi3 is
implemented.

torsion_order( self)

Return the order of the torsion subgroup.

sage: e = EllipticCurve('11a')
sage: e.torsion_order()
5
sage: type(e.torsion_order())
<type 'sage.rings.integer.Integer'>
sage: e = EllipticCurve([1,2,3,4,5])
sage: e.torsion_order()
1
sage: type(e.torsion_order())
<type 'sage.rings.integer.Integer'>

torsion_subgroup( self, [flag=0])

Returns the torsion subgroup of this elliptic curve.

Input:

flag
- (default: 0) chooses PARI algorithm: flag = 0: uses Doud algorithm flag = 1: uses Lutz-Nagell algorithm

Output: The EllipticCurveTorsionSubgroup instance associated to this elliptic curve.

sage: EllipticCurve('11a').torsion_subgroup()
Torsion Subgroup isomorphic to Multiplicative Abelian Group isomorphic to
C5 associated to the Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x -
20 over Rational Field
sage: EllipticCurve('37b').torsion_subgroup()
Torsion Subgroup isomorphic to Multiplicative Abelian Group isomorphic to
C3 associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x -
50 over Rational Field

sage: e = EllipticCurve([-1386747,368636886]);e
Elliptic Curve defined by y^2  = x^3 - 1386747*x + 368636886 over Rational
Field
sage: G = e.torsion_subgroup(); G
Torsion Subgroup isomorphic to Multiplicative Abelian
Group isomorphic to C8 x C2 associated to the Elliptic
Curve defined by y^2 = x^3 - 1386747*x + 368636886 over
Rational Field
sage: G.0
(1227 : 22680 : 1)
sage: G.1
(282 : 0 : 1)
sage: list(G)
[1, P1, P0, P0*P1, P0^2, P0^2*P1, P0^3, P0^3*P1, P0^4, P0^4*P1, P0^5,
P0^5*P1, P0^6, P0^6*P1, P0^7, P0^7*P1]

two_descent( self, [verbose=True], [selmer_only=False], [first_limit=20], [second_limit=8], [n_aux=-1], [second_descent=1])

Compute 2-descent data for this curve.

Input:

verbose
- (default: True) print what mwrank is doing If False, *no output* is
selmer_only
- (default: False) selmer_only switch
first_limit
- (default: 20) firstlim is bound on |x|+|z| second_limit- (default: 8) secondlim is bound on log max |x|,|z| , i.e. logarithmic
n_aux
- (default: -1) n_aux only relevant for general 2-descent when 2-torsion trivial; n_aux=-1 causes default to be used (depends on method)
second_descent
- (default: True) second_descent only relevant for descent via 2-isogeny
Output:

Nothing - nothing is returned (though much is printed unless verbose=False)

sage: E=EllipticCurve('37a1')
sage: E.two_descent(verbose=False) # no output

two_descent_simon( self, [verbose=0], [lim1=5], [lim3=50], [limtriv=10], [maxprob=20], [limbigprime=30])

Given a curve with no 2-torsion, computes (probably) the rank of the Mordell-Weil group, with certainty the rank of the 2-Selmer group, and a list of independent points on the curve.

Input:

verbose
- integer, 0,1,2,3; (default: 0), the verbosity level
lim1
- (default: 5) limite des points triviaux sur les quartiques
lim3
- (default: 50) limite des points sur les quartiques ELS
limtriv
- (default: 10) limite des points triviaux sur la courbe elliptique
maxprob
- (default: 20)
limbigprime
- (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don't any probabilistic tests.

Output:
integer
- "probably" the rank of self
integer
- the 2-rank of the Selmer group
list
- list of independent points on the curve.

IMPLEMENTATION: Uses Denis Simon's GP/PARI scripts from http://www.math.unicaen.fr/~simon/

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

We compute the ranks of the curves of lowest known conductor up to rank $ 8$ . Amazingly, each of these computations finishes almost instantly!

sage: E = EllipticCurve('11a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(0, 0, [])
sage: E = EllipticCurve('37a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(1, 1, [(0 : 0 : 1)])
sage: E = EllipticCurve('389a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)])
sage: E = EllipticCurve('5077a1')
sage: set_random_seed(0)
sage: E.simon_two_descent()
(3, 3, [(1 : 0 : 1), (2 : -1 : 1), (0 : 2 : 1)])

In this example Simon's program does not find any points, though it does correctly compute the rank of the 2-Selmer group.

sage: E = EllipticCurve([1, -1, 0, -751055859, -7922219731979])     # long (0.6 seconds)
sage: set_random_seed(0)
sage: E.simon_two_descent ()
(1, 1, [])

The rest of these entries were taken from Tom Womack's page http://tom.womack.net/maths/conductors.htm

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: set_random_seed(0)
sage: E.simon_two_descent()
(4, 4, [(6 : -5 : 1), (4 : 3 : 1), (5 : -3 : 1), (8 : -15 : 1)])
sage: E = EllipticCurve([0, 0, 1, -79, 342])
sage: set_random_seed(0)
sage: E.simon_two_descent()
(5, 5, [(5 : 8 : 1), (10 : 23 : 1), (3 : 11 : 1), (4 : -10 : 1), (0 : 18 :
1)])
sage: E = EllipticCurve([1, 1, 0, -2582, 48720])
sage: set_random_seed(0)
sage: r, s, G = E.simon_two_descent(); r,s
(6, 6)
sage: E = EllipticCurve([0, 0, 0, -10012, 346900])
sage: set_random_seed(0)
sage: r, s, G = E.simon_two_descent(); r,s
(7, 7)
sage: E = EllipticCurve([0, 0, 1, -23737, 960366])    
sage: set_random_seed(0)
sage: r, s, G = E.simon_two_descent(); r,s
(8, 8)

two_torsion_rank( self)

Return the dimension of the 2-torsion subgroup of $ E(\mathbf{Q})$ .

This will be 0, 1 or 2.

NOTE: As s side-effect of calling this function, the full torsion subgroup of the curve is computed (if not already cached). A simpler implementation of this function would be possible (by counting the roots of the 2-division polynomial), but the full torsion subgroup computation is not expensive.

sage: EllipticCurve('11a1').two_torsion_rank()
0
sage: EllipticCurve('14a1').two_torsion_rank()
1
sage: EllipticCurve('15a1').two_torsion_rank()
2

Special Functions: __init__,$ \,$ _EllipticCurve_rational_field__adjust_heegner_index,$ \,$ _is_surjective,$ \,$ _multiple_of_degree_of_isogeny_to_optimal_curve,$ \,$ _set_conductor,$ \,$ _set_cremona_label,$ \,$ _set_gens,$ \,$ _set_modular_degree,$ \,$ _set_rank,$ \,$ _set_torsion_order

_EllipticCurve_rational_field__adjust_heegner_index( self, a)

Take the square root of the interval that contains the Heegner index.

sage: E = EllipticCurve('11a1')
sage: a = RIF(sqrt(2))-1.4142135623730951
sage: E._EllipticCurve_rational_field__adjust_heegner_index(a)
[0.0000000... .. 1.490116...e-8]

_multiple_of_degree_of_isogeny_to_optimal_curve( self)

Internal function returning an integer m such that the degree of the isogeny between this curve and the optimal curve in its isogeny class is a divisor of m.

sage: E=EllipticCurve('11a1')
sage: E._multiple_of_degree_of_isogeny_to_optimal_curve()
25
sage: E=EllipticCurve('11a2')
sage: E._multiple_of_degree_of_isogeny_to_optimal_curve()
5
sage: E=EllipticCurve('11a3')
sage: E._multiple_of_degree_of_isogeny_to_optimal_curve()
5

_set_conductor( self, N)

Internal function to set the cached conductor of this elliptic curve to N.

WARNING: No checking is done! Not intended for use by users. Setting to the wrong value will cause strange problems (see examples).

sage: E=EllipticCurve('37a1')
sage: E._set_conductor(99)      # bogus value -- not checked
sage: E.conductor()             # returns bogus cached value
99

This will not work since the conductor is used when searching the database:

sage: E._set_conductor(E.database_curve().conductor()) 
Traceback (most recent call last): 
...  
RuntimeError: Elliptic curve ... not in the database.
sage: E._set_conductor(EllipticCurve(E.a_invariants()).database_curve().conductor()) 
sage: E.conductor()             # returns correct value
37

_set_cremona_label( self, L)

Internal function to set the cached label of this elliptic curve to L.

WARNING: No checking is done! Not intended for use by users.

sage: E=EllipticCurve('37a1')
sage: E._set_cremona_label('bogus')
sage: E.label()
'bogus'
sage: E.database_curve().label()
'37a1'
sage: E.label() # no change
'bogus'
sage: E._set_cremona_label(E.database_curve().label())
sage: E.label() # now it is correct
'37a1'

_set_gens( self, gens)

Internal function to set the cached generators of this elliptic curve to gens.

WARNING: No checking is done!

sage: E=EllipticCurve('5077a1')
sage: E.rank()
3
sage: E.gens() # random
[(-2 : 3 : 1), (-7/4 : 25/8 : 1), (1 : -1 : 1)]
sage: E._set_gens([]) # bogus list
sage: E.rank()        # unchanged 
3
sage: E._set_gens(E.database_curve().gens())
sage: E.gens()
[(-2 : 3 : 1), (-7/4 : 25/8 : 1), (1 : -1 : 1)]

_set_modular_degree( self, deg)

Internal function to set the cached modular degree of this elliptic curve to deg.

WARNING: No checking is done!

sage: E=EllipticCurve('5077a1')
sage: E.modular_degree()
1984
sage: E._set_modular_degree(123456789)
sage: E.modular_degree()
123456789
sage: E._set_modular_degree(E.database_curve().modular_degree())
sage: E.modular_degree()
1984

_set_rank( self, r)

Internal function to set the cached rank of this elliptic curve to r.

WARNING: No checking is done! Not intended for use by users.

sage: E=EllipticCurve('37a1')
sage: E._set_rank(99)  # bogus value -- not checked
sage: E.rank()         # returns bogus cached value
99
sage: E.gens()         # causes actual rank to be computed
[(0 : -1 : 1)]
sage: E.rank()         # the correct rank
1

_set_torsion_order( self, t)

Internal function to set the cached torsion order of this elliptic curve to t.

WARNING: No checking is done! Not intended for use by users.

sage: E=EllipticCurve('37a1')
sage: E._set_torsion_order(99)  # bogus value -- not checked
sage: E.torsion_order()         # returns bogus cached value
99
sage: T = E.torsion_subgroup()  # causes actual torsion to be computed
sage: E.torsion_order()         # the correct value
1

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