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def kolyvagin_point(self, D, c=ZZ(1), check=True):
"\n Return the Kolyvagin point on this curve associated to the\n quadratic imaginary field `K=\\QQ(\\sqrt{D})` and conductor `c`.\n\n INPUT:\n\n - `D` -- a Heegner discriminant\n\n - `c` -- (default: 1) conductor, must be coprime to `... |
def ell_heegner_discriminants(self, bound):
"\n Return the list of self's Heegner discriminants between -1 and\n -bound.\n\n INPUT:\n\n - ``bound (int)`` -- upper bound for -discriminant\n\n OUTPUT: The list of Heegner discriminants between -1 and -bound for\n the given elliptic curve.\n\n EX... |
def ell_heegner_discriminants_list(self, n):
"\n Return the list of self's first `n` Heegner discriminants smaller\n than -5.\n\n INPUT:\n\n - ``n (int)`` -- the number of discriminants to compute\n\n OUTPUT: The list of the first n Heegner discriminants smaller than\n -5 for the given elliptic ... |
def heegner_point_height(self, D, prec=2, check_rank=True):
"\n Use the Gross-Zagier formula to compute the Neron-Tate canonical\n height over `K` of the Heegner point corresponding to `D`, as an\n interval (it is computed to some precision using `L`-functions).\n\n If the curve has rank at least 2, t... |
def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, verbose_mwrank=False, check_rank=True):
"\n Return an interval that contains the index of the Heegner\n point `y_K` in the group of `K`-rational points modulo torsion\n on this elliptic curve, computed using the Gross-Zagier\n formul... |
def _adjust_heegner_index(self, a):
"\n Take the square root of the interval that contains the Heegner\n index.\n\n EXAMPLES::\n\n sage: E = EllipticCurve('11a1')\n sage: a = RIF(sqrt(2)) - RIF(1.4142135623730951)\n sage: E._adjust_heegner_index(a)\n 1.?e-8\n "
if (a.lo... |
def heegner_index_bound(self, D=0, prec=5, max_height=None):
"\n Assume ``self`` has rank 0.\n\n Return a list `v` of primes such that if an odd prime `p` divides\n the index of the Heegner point in the group of rational points\n modulo torsion, then `p` is in `v`.\n\n If 0 is in the interval of th... |
def _heegner_index_in_EK(self, D):
"\n Return the index of the sum of `E(\\QQ)/tor + E^D(\\QQ)/tor` in `E(K)/tor`.\n\n INPUT:\n\n - `D` -- negative integer; the Heegner discriminant\n\n OUTPUT: A power of 2 -- the given index.\n\n EXAMPLES:\n\n We compute the index for a rank 2 curve and found t... |
def heegner_sha_an(self, D, prec=53):
"\n Return the conjectural (analytic) order of Sha for E over the field `K=\\QQ(\\sqrt{D})`.\n\n INPUT:\n\n - `D` -- negative integer; the Heegner discriminant\n\n - prec -- integer (default: 53); bits of precision to\n compute analytic order of Sha\n\n OU... |
def _heegner_forms_list(self, D, beta=None, expected_count=None):
"\n Returns a list of quadratic forms corresponding to Heegner points\n with discriminant `D` and a choice of `\\beta` a square root of\n `D` mod `4N`. Specifically, given a quadratic form\n `f = Ax^2 + Bxy + Cy^2` we let `\\tau_f` be a... |
def _heegner_best_tau(self, D, prec=None):
"\n Given a discriminant `D`, find the Heegner point `\\tau` in the\n upper half plane with largest imaginary part (which is optimal\n for evaluating the modular parametrization). If the optional\n parameter ``prec`` is given, return `\\tau` to ``prec`` bits ... |
def satisfies_heegner_hypothesis(self, D):
"\n Returns ``True`` precisely when `D` is a fundamental discriminant that\n satisfies the Heegner hypothesis for this elliptic curve.\n\n EXAMPLES::\n\n sage: E = EllipticCurve('11a1')\n sage: E.satisfies_heegner_hypothesis(-7)\n True\n ... |
class UnionOfIntervals():
'\n A class representing a finite union of closed intervals in\n `\\RR` which can be scaled, shifted, intersected, etc.\n\n The intervals are represented as an ordered list of their\n endpoints, which may include `-\\infty` and `+\\infty`.\n\n EXAMPLES::\n\n sage: f... |
def nonneg_region(f):
'\n Returns the UnionOfIntervals representing the region where ``f`` is non-negative.\n\n INPUT:\n\n - ``f`` (polynomial) -- a univariate polynomial over `\\RR`.\n\n OUTPUT:\n\n A UnionOfIntervals representing the set `\\{x \\in\\RR mid f(x) \\ge 0\\}`.\n\n EXAMPLES::\n\n ... |
def inf_max_abs(f, g, D):
"\n Returns `\\inf_D(\\max(|f|, |g|))`.\n\n INPUT:\n\n - ``f``, ``g`` (polynomials) -- real univariate polynomials\n\n - ``D`` (:class:`UnionOfIntervals`) -- a subset of `\\RR`\n\n OUTPUT:\n\n A real number approximating the value of `\\inf_D(\\max(|f|, |g|))`.\n\n A... |
def min_on_disk(f, tol, max_iter=10000):
'\n Returns the minimum of a real-valued complex function on a square.\n\n INPUT:\n\n - ``f`` -- a function from CIF to RIF\n\n - ``tol`` (real) -- a positive real number\n\n - ``max_iter`` (integer, default 10000) -- a positive integer\n bounding the n... |
def rat_term_CIF(z, try_strict=True):
'\n Compute the value of `u/(1-u)^2` in ``CIF``, where `u=\\exp(2\\pi i z)`.\n\n INPUT:\n\n - ``z`` (complex) -- a CIF element\n\n - ``try_strict`` (bool) -- flag\n\n EXAMPLES::\n\n sage: from sage.schemes.elliptic_curves.height import rat_term_CIF\n ... |
def eps(err, is_real):
'\n Return a Real or Complex interval centered on 0 with radius err.\n\n INPUT:\n\n - ``err`` (real) -- a positive real number, the radius of the interval\n\n - ``is_real`` (boolean) -- if True, returns a real interval in\n RIF, else a complex interval in CIF\n\n OUTPUT:... |
class EllipticCurveCanonicalHeight():
'\n Class for computing canonical heights of points on elliptic curves\n defined over number fields, including rigorous lower bounds for\n the canonical height of non-torsion points.\n\n EXAMPLES::\n\n sage: from sage.schemes.elliptic_curves.height import E... |
class EllipticCurveHom(Morphism):
'\n Base class for elliptic-curve morphisms.\n '
def __init__(self, *args, **kwds):
"\n Constructor for elliptic-curve morphisms.\n\n EXAMPLES::\n\n sage: E = EllipticCurve(GF(257^2), [5,5])\n sage: P = E.lift_x(1)\n ... |
def compare_via_evaluation(left, right):
"\n Test if two elliptic-curve morphisms are equal by evaluating\n them at enough points.\n\n INPUT:\n\n - ``left``, ``right`` -- :class:`EllipticCurveHom` objects\n\n ALGORITHM:\n\n We use the fact that two isogenies of equal degree `d` must be\n the ... |
def find_post_isomorphism(phi, psi):
"\n Given two isogenies `\\phi: E\\to E'` and `\\psi: E\\to E''`\n which are equal up to post-isomorphism defined over the\n same field, find that isomorphism.\n\n In other words, this function computes an isomorphism\n `\\alpha: E'\\to E''` such that `\\alpha\\... |
def compute_trace_generic(phi):
"\n Compute the trace of the given elliptic-curve endomorphism.\n\n ALGORITHM: Simple variant of Schoof's algorithm.\n For enough small primes `\\ell`, we find an order-`\\ell` point `P`\n on `E` and use a discrete-logarithm calculation to find the unique\n scalar `t... |
def _eval_factored_isogeny(phis, P):
'\n This method pushes a point `P` through a given sequence ``phis``\n of compatible isogenies.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\n sage: from sage.schemes.elliptic_curves import hom_composite\n sage: E = EllipticCurve(GF(41... |
def _compute_factored_isogeny_prime_power(P, l, n, split=0.8):
'\n This method takes a point `P` of order `\\ell^n` and returns\n a sequence of degree-`\\ell` isogenies whose composition has\n the subgroup generated by `P` as its kernel.\n\n The optional argument ``split``, a real number between\n ... |
def _compute_factored_isogeny_single_generator(P):
'\n This method takes a point `P` and returns a sequence of\n prime-degree isogenies whose composition has the subgroup\n generated by `P` as its kernel.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\n sage: from sage.schemes.... |
def _compute_factored_isogeny(kernel):
'\n This method takes a set of points on an elliptic curve\n and returns a sequence of isogenies whose composition\n has the subgroup generated by that subset as its kernel.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\n sage: from sage.... |
class EllipticCurveHom_composite(EllipticCurveHom):
_degree = None
_phis = None
def __init__(self, E, kernel, codomain=None, model=None):
"\n Construct a composite isogeny with given kernel (and optionally,\n prescribed codomain curve). The isogeny is decomposed into steps\n ... |
class EllipticCurveHom_frobenius(EllipticCurveHom):
_degree = None
def __init__(self, E, power=1):
"\n Construct a Frobenius isogeny on a given curve with a given\n power of the base-ring characteristic.\n\n Writing `n` for the parameter ``power`` (default: `1`), the\n iso... |
class EllipticCurveHom_scalar(EllipticCurveHom):
def __init__(self, E, m):
'\n Construct a scalar-multiplication map on an elliptic curve.\n\n TESTS::\n\n sage: from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar\n sage: E = EllipticCurve([1,1])... |
class EllipticCurveHom_sum(EllipticCurveHom):
_degree = None
_phis = None
def __init__(self, phis, domain=None, codomain=None):
'\n Construct a sum morphism of elliptic curves from its summands.\n (For empty sums, the domain and codomain curves must be given.)\n\n EXAMPLES::\... |
def _choose_IJK(n):
'\n Helper function to choose an "index system" for the set\n `\\{1,3,5,7,...,n-2\\}` where `n \\geq 5` is an odd integer.\n\n INPUT:\n\n - ``n`` -- odd :class:`~sage.rings.integer.Integer` `\\geq 5`\n\n REFERENCES: [BDLS2020]_, Examples 4.7 and 4.12\n\n EXAMPLES::\n\n ... |
def _points_range(rr, P, Q=None):
'\n Return an iterator yielding all points `Q + [i]P` where `i` runs\n through the :class:`range` object ``rr``.\n\n INPUT:\n\n - ``rr`` -- :class:`range` object defining a sequence `S \\subseteq \\ZZ`\n - ``P`` -- element of an additive abelian group\n - ``Q`` ... |
class FastEllipticPolynomial():
"\n A class to represent and evaluate an *elliptic polynomial*,\n and optionally its derivative, in essentially square-root time.\n\n The elliptic polynomials computed by this class are of the form\n\n .. MATH::\n\n h_S(Z) = \\prod_{i\\in S} (Z - x(Q + [i]P))\n\n... |
def _point_outside_subgroup(P):
'\n Simple helper function to return a point on an elliptic\n curve `E` that is not a multiple of a given point `P`.\n The base field is extended if (and only if) necessary.\n\n INPUT:\n\n - ``P`` -- a point on an elliptic curve over a finite field\n\n EXAMPLES::\... |
class EllipticCurveHom_velusqrt(EllipticCurveHom):
'\n This class implements separable odd-degree isogenies of elliptic\n curves over finite fields using the square-root Vélu algorithm.\n\n The complexity is `\\tilde O(\\sqrt{\\ell})` base-field operations,\n where `\\ell` is the degree.\n\n REFERE... |
def _random_example_for_testing():
'\n Function to generate somewhat random valid Vélu inputs\n for testing purposes.\n\n EXAMPLES::\n\n sage: from sage.schemes.elliptic_curves.hom_velusqrt import _random_example_for_testing\n sage: E, K = _random_example_for_testing()\n sage: E ... |
@richcmp_method
class IsogenyClass_EC(SageObject):
'\n Isogeny class of an elliptic curve.\n\n .. NOTE::\n\n The current implementation chooses a curve from each isomorphism\n class in the isogeny class. Over `\\QQ` this is a unique reduced\n minimal model in each isomorphism class. Ov... |
class IsogenyClass_EC_NumberField(IsogenyClass_EC):
'\n Isogeny classes for elliptic curves over number fields.\n '
def __init__(self, E, reducible_primes=None, algorithm='Billerey', minimal_models=True):
"\n INPUT:\n\n - ``E`` -- an elliptic curve over a number field.\n\n ... |
class IsogenyClass_EC_Rational(IsogenyClass_EC_NumberField):
'\n Isogeny classes for elliptic curves over `\\QQ`.\n '
def __init__(self, E, algorithm='sage', label=None, empty=False):
'\n INPUT:\n\n - ``E`` -- an elliptic curve over `\\QQ`.\n\n - ``algorithm`` -- a string (... |
def isogeny_degrees_cm(E, verbose=False):
"\n Return a list of primes `\\ell` sufficient to generate the\n isogeny class of `E`, where `E` has CM.\n\n INPUT:\n\n - ``E`` -- An elliptic curve defined over a number field.\n\n OUTPUT:\n\n A finite list of primes `\\ell` such that every curve isogen... |
def possible_isogeny_degrees(E, algorithm='Billerey', max_l=None, num_l=None, exact=True, verbose=False):
"\n Return a list of primes `\\ell` sufficient to generate the\n isogeny class of `E`.\n\n INPUT:\n\n - ``E`` -- An elliptic curve defined over a number field.\n\n - ``algorithm`` (string, defa... |
@cached_function
def Fricke_polynomial(l):
'\n Fricke polynomial for ``l`` =2,3,5,7,13.\n\n For these primes (and these only) the modular curve `X_0(l)` has\n genus zero, and its field is generated by a single modular\n function called the Fricke module (or Hauptmodul), `t`. There is\n a classical... |
@cached_function
def Fricke_module(l):
'\n Fricke module for ``l`` =2,3,5,7,13.\n\n For these primes (and these only) the modular curve `X_0(l)` has\n genus zero, and its field is generated by a single modular\n function called the Fricke module (or Hauptmodul), `t`. There is\n a classical choice ... |
@cached_function
def Psi(l, use_stored=True):
'\n Generic kernel polynomial for genus zero primes.\n\n For each of the primes `l` for which `X_0(l)` has genus zero\n (namely `l=2,3,5,7,13`), we may define an elliptic curve `E_t`\n over `\\QQ(t)`, with coefficients in `\\ZZ[t]`, which has good\n red... |
def isogenies_prime_degree_genus_0(E, l=None, minimal_models=True):
"\n Return list of ``l`` -isogenies with domain ``E``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``l`` -- either None or 2, 3, 5, 7, or 13.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curve... |
@cached_function
def _sporadic_Q_data(j):
"\n Return technical data used in computing sporadic isogenies over `\\QQ`.\n\n INPUT:\n\n - ``j`` -- The `j`-invariant of a sporadic curve, i.e. one of the\n keys of ``sporadic_j``.\n\n OUTPUT:\n\n ``([a4,a6],coeffs)`` where ``[a4,a6]`` are the coeffi... |
def isogenies_sporadic_Q(E, l=None, minimal_models=True):
"\n Return a list of sporadic l-isogenies from E (l = 11, 17, 19, 37,\n 43, 67 or 163). Only for elliptic curves over `\\QQ`.\n\n INPUT:\n\n - ``E`` -- an elliptic curve defined over `\\QQ`.\n\n - ``l`` -- either None or a prime number.\n\n ... |
def isogenies_2(E, minimal_models=True):
"\n Return a list of all 2-isogenies with domain ``E``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or semi-minimal models. Over\n fields of larg... |
def isogenies_3(E, minimal_models=True):
"\n Return a list of all 3-isogenies with domain ``E``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or semi-minimal models. Over\n fields of larg... |
def isogenies_5_0(E, minimal_models=True):
"\n Return a list of all the 5-isogenies with domain ``E`` when the\n j-invariant is 0.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 0.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be min... |
def isogenies_5_1728(E, minimal_models=True):
"\n Return a list of 5-isogenies with domain ``E`` when the j-invariant is\n 1728.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 1728.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be mi... |
def isogenies_7_0(E, minimal_models=True):
"\n Return list of all 7-isogenies from E when the j-invariant is 0.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 0.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or semi-minimal m... |
def isogenies_7_1728(E, minimal_models=True):
"\n Return list of all 7-isogenies from E when the j-invariant is 1728.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 1728.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or semi-... |
def isogenies_13_0(E, minimal_models=True):
"\n Return list of all 13-isogenies from E when the j-invariant is 0.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 0.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or semi-minimal... |
def isogenies_13_1728(E, minimal_models=True):
'\n Return list of all 13-isogenies from E when the j-invariant is 1728.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 1728.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or sem... |
@cached_function
def _hyperelliptic_isogeny_data(l):
"\n Helper function for elliptic curve isogenies.\n\n INPUT:\n\n - ``l`` -- a prime in [11, 17, 19, 23, 29, 31, 41, 47, 59, 71]\n\n OUTPUT:\n\n - A dict holding a collection of precomputed data needed for computing `l`-isogenies.\n\n EXAMPLES:... |
@cached_function
def Psi2(l):
'\n Return the generic kernel polynomial for hyperelliptic `l`-isogenies.\n\n INPUT:\n\n - ``l`` -- either 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.\n\n OUTPUT:\n\n The generic `l`-kernel polynomial.\n\n EXAMPLES::\n\n sage: from sage.schemes.elliptic_curves... |
def isogenies_prime_degree_genus_plus_0(E, l=None, minimal_models=True):
"\n Return list of ``l`` -isogenies with domain ``E``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``l`` -- either None or 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.\n\n - ``minimal_models`` (bool, default ``True``) -- ... |
def isogenies_prime_degree_genus_plus_0_j0(E, l, minimal_models=True):
"\n Return a list of hyperelliptic ``l`` -isogenies with domain ``E`` when `j(E)=0`.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 0.\n\n - ``l`` -- 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.\n\n - ``minimal_model... |
def isogenies_prime_degree_genus_plus_0_j1728(E, l, minimal_models=True):
"\n Return a list of ``l`` -isogenies with domain ``E`` when `j(E)=1728`.\n\n INPUT:\n\n - ``E`` -- an elliptic curve with j-invariant 1728.\n\n - ``l`` -- 11, 17, 19, 23, 29, 31, 41, 47, 59, or 71.\n\n - ``minimal_models`` (... |
@cached_function
def _least_semi_primitive(p):
'\n Return the smallest semi-primitive root modulo `p`, i.e., generator of the group `(\\ZZ/p\\ZZ)^*/\\{1,-1\\}`.\n\n INPUT:\n\n - ``p`` -- an odd prime power.\n\n OUTPUT:\n\n the smallest semi-primitive root modulo `p`.\n\n .. NOTE::\n\n Thi... |
def is_kernel_polynomial(E, m, f):
'\n Test whether ``E`` has a cyclic isogeny of degree ``m`` with kernel\n polynomial ``f``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``m`` -- a positive integer.\n\n - ``f`` -- a polynomial over the base field of ``E``.\n\n OUTPUT:\n\n (bool) ``... |
def isogenies_prime_degree_general(E, l, minimal_models=True):
"\n Return all separable ``l``-isogenies with domain ``E``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``l`` -- a prime.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minima... |
def isogenies_prime_degree(E, l, minimal_models=True):
"\n Return all separable ``l``-isogenies with domain ``E``.\n\n INPUT:\n\n - ``E`` -- an elliptic curve.\n\n - ``l`` -- a prime.\n\n - ``minimal_models`` (bool, default ``True``) -- if ``True``, all\n curves computed will be minimal or sem... |
def Jacobian(X, **kwds):
'\n Return the Jacobian.\n\n INPUT:\n\n - ``X`` -- polynomial, algebraic variety, or anything else that\n has a Jacobian elliptic curve.\n\n - ``kwds`` -- optional keyword arguments.\n\n The input ``X`` can be one of the following:\n\n * A polynomial, see :func:`Jac... |
def Jacobian_of_curve(curve, morphism=False):
'\n Return the Jacobian of a genus-one curve\n\n INPUT:\n\n - ``curve`` -- a one-dimensional algebraic variety of genus one.\n\n OUTPUT: Its Jacobian elliptic curve.\n\n EXAMPLES::\n\n sage: R.<u,v,w> = QQ[]\n sage: C = Curve(u^3 + v^3 + w... |
def Jacobian_of_equation(polynomial, variables=None, curve=None):
'\n Construct the Jacobian of a genus-one curve given by a polynomial.\n\n INPUT:\n\n - ``F`` -- a polynomial defining a plane curve of genus one. May\n be homogeneous or inhomogeneous.\n\n - ``variables`` -- list of two or three v... |
@richcmp_method
class KodairaSymbol_class(SageObject):
'\n Class to hold a Kodaira symbol of an elliptic curve over a\n `p`-adic local field.\n\n Users should use the ``KodairaSymbol()`` function to construct\n Kodaira Symbols rather than use the class constructor directly.\n '
def __init__(se... |
def KodairaSymbol(symbol):
'\n Return the specified Kodaira symbol.\n\n INPUT:\n\n - ``symbol`` (string or integer) -- Either a string of the form "I0", "I1", ..., "In", "II", "III", "IV", "I0*", "I1*", ..., "In*", "II*", "III*", or "IV*", or an integer encoding a Kodaira symbol using PARI\'s conventions... |
def c4c6_nonsingular(c4, c6):
"\n Check if c4, c6 are integral with valid associated discriminant.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n OUTPUT:\n\n Boolean, True if c4, c6 are both integral and c4^3-c6^2 is a\n nonzero multiple of 1728.\n\n EXAMPLES:\n\n Over `... |
def c4c6_model(c4, c6, assume_nonsingular=False):
"\n Return the elliptic curve [0,0,0,-c4/48,-c6/864] with given c-invariants.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``assume_nonsingular`` (boolean, default False) -- if True,\n check for integrality and nosingularity... |
def make_integral(a, P, e):
"\n Returns b in O_K with P^e|(a-b), given a in O_{K,P}.\n\n INPUT:\n\n - ``a`` -- a number field element integral at ``P``\n\n - ``P`` -- a prime ideal of the number field\n\n - ``e`` -- a positive integer\n\n OUTPUT:\n\n A globally integral number field element `... |
def sqrt_mod_4(x, P):
"\n Returns a local square root mod 4, if it exists.\n\n INPUT:\n\n - ``x`` -- an integral number field element\n\n - ``P`` -- a prime ideal of the number field dividing 2\n\n OUTPUT:\n\n A pair (True, r) where that `r^2-x` has valuation at least `2e`,\n or (False, 0) if... |
def test_b2_local(c4, c6, P, b2, debug=False):
"\n Test if b2 gives a valid model at a prime dividing 3.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``P`` -- a prime ideal of the number field which divides 3\n\n - ``b2`` -- an element of the number field\n\n OUTPUT:\n\n ... |
def test_b2_global(c4, c6, b2, debug=False):
"\n Test if b2 gives a valid model at all primes dividing 3.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``b2`` -- an element of the number field\n\n OUTPUT:\n\n The elliptic curve which is the (b2/12,0,0)-transform of\n [0,... |
def check_Kraus_local_3(c4, c6, P, assume_nonsingular=False, debug=False):
"\n Test if c4,c6 satisfy Kraus's conditions at a prime P dividing 3.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``P`` -- a prime ideal of the number field which divides 3\n\n - ``assume_nonsingular`... |
def test_a1a3_local(c4, c6, P, a1, a3, debug=False):
"\n Test if a1,a3 are valid at a prime P dividing 2.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``P`` -- a prime ideal of the number field which divides 2\n\n - ``a1``, ``a3`` -- elements of the number field\n\n OUTPUT... |
def test_a1a3_global(c4, c6, a1, a3, debug=False):
"\n Test if a1,a3 are valid at all primes P dividing 2.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``a1``, ``a3`` -- elements of the number field\n\n OUTPUT:\n\n The elliptic curve which is the (a1^2/12,a1/2,a3/2)-transf... |
def test_rst_global(c4, c6, r, s, t, debug=False):
"\n Test if the (r,s,t)-transform of the standard c4,c6-model is integral.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``r``, ``s``, ``t`` -- elements of the number field\n\n OUTPUT:\n\n The elliptic curve which is the (r... |
def check_Kraus_local_2(c4, c6, P, a1=None, assume_nonsingular=False):
"\n Test if c4,c6 satisfy Kraus's conditions at a prime P dividing 2.\n\n INPUT:\n\n - ``c4``, ``c6`` -- integral elements of a number field\n\n - ``P`` -- a prime ideal of the number field which divides 2\n\n - ``a1`` -- an int... |
def check_Kraus_local(c4, c6, P, assume_nonsingular=False):
"\n Check Kraus's conditions locally at a prime P.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``P`` -- a prime ideal of the number field\n\n - ``assume_nonsingular`` (boolean, default False) -- if True,\n chec... |
def check_Kraus_global(c4, c6, assume_nonsingular=False, debug=False):
"\n Test if c4,c6 satisfy Kraus's conditions at all primes.\n\n INPUT:\n\n - ``c4``, ``c6`` -- elements of a number field\n\n - ``assume_nonsingular`` (boolean, default False) -- if True,\n check for integrality and nosingular... |
def semi_global_minimal_model(E, debug=False):
"\n Return a global minimal model for this elliptic curve if it\n exists, or a model minimal at all but one prime otherwise.\n\n INPUT:\n\n - ``E`` -- an elliptic curve over a number field\n\n - ``debug`` (boolean, default ``False``) -- if ``True``, pr... |
class Lseries_ell(SageObject):
'\n An elliptic curve `L`-series.\n '
def __init__(self, E):
'\n Create an elliptic curve `L`-series.\n\n EXAMPLES::\n\n sage: EllipticCurve([1..5]).lseries()\n Complex L-series of the Elliptic Curve\n defined by y^2... |
def mod5family(a, b):
'\n Formulas for computing the family of elliptic curves with\n congruent mod-5 representation.\n\n EXAMPLES::\n\n sage: from sage.schemes.elliptic_curves.mod5family import mod5family\n sage: mod5family(0,1)\n Elliptic Curve defined by y^2 = x^3 + (t^30+30*t^29+... |
def classical_modular_polynomial(l, j=None):
'\n Return the classical modular polynomial `\\Phi_\\ell`, either as a\n "generic" bivariate polynomial over `\\ZZ`, or as an "instantiated"\n modular polynomial where one variable has been replaced by the\n given `j`-invariant.\n\n Generic polynomials a... |
def _set_cache_bound(bnd):
'\n Internal helper function to allow setting the caching cutoff for\n :func:`classical_modular_polynomial`.\n\n Exposed as ``classical_modular_polynomial.set_cache_bound()``.\n\n EXAMPLES::\n\n sage: import sage.schemes.elliptic_curves.mod_poly as m\n sage: m.... |
class ModularParameterization():
"\n This class represents the modular parametrization of an elliptic curve\n\n .. MATH::\n\n \\phi_E: X_0(N) \\rightarrow E.\n\n Evaluation is done by passing through the lattice representation of `E`.\n\n EXAMPLES::\n\n sage: phi = EllipticCurve('11a1').... |
@richcmp_method
class pAdicLseries(SageObject):
"\n The `p`-adic L-series of an elliptic curve.\n\n EXAMPLES:\n\n An ordinary example::\n\n sage: e = EllipticCurve('389a')\n sage: L = e.padic_lseries(5)\n sage: L.series(0)\n Traceback (most recent call last):\n ...\n ... |
class pAdicLseriesOrdinary(pAdicLseries):
def series(self, n=2, quadratic_twist=(+ 1), prec=5, eta=0):
'\n Return the `n`-th approximation to the `p`-adic L-series, in the\n component corresponding to the `\\eta`-th power of the Teichmueller\n character, as a power series in `T` (cor... |
class pAdicLseriesSupersingular(pAdicLseries):
def series(self, n=3, quadratic_twist=(+ 1), prec=5, eta=0):
'\n Return the `n`-th approximation to the `p`-adic L-series as a\n power series in `T` (corresponding to `\\gamma-1` with\n `\\gamma=1+p` as a generator of `1+p\\ZZ_p`). Each... |
def __check_padic_hypotheses(self, p):
"\n Helper function that determines if `p`\n is an odd prime of good ordinary reduction.\n\n EXAMPLES::\n\n sage: E = EllipticCurve('11a1')\n sage: from sage.schemes.elliptic_curves.padics import __check_padic_hypotheses\n sage: __check_padic_hy... |
def _normalize_padic_lseries(self, p, normalize, implementation, precision):
'\n Normalize parameters for :meth:`padic_lseries`.\n\n TESTS::\n\n sage: from sage.schemes.elliptic_curves.padics import _normalize_padic_lseries\n sage: u = _normalize_padic_lseries(None, 5, None, \'sage\', 10)\n ... |
@cached_method(key=_normalize_padic_lseries)
def padic_lseries(self, p, normalize=None, implementation='eclib', precision=None):
'\n Return the `p`-adic `L`-series of self at\n `p`, which is an object whose approx method computes\n approximation to the true `p`-adic `L`-series to\n any desired precisi... |
def padic_regulator(self, p, prec=20, height=None, check_hypotheses=True):
'\n Compute the cyclotomic `p`-adic regulator of this curve.\n The model of the curve needs to be integral and minimal at `p`.\n Moreover the reduction at `p` should not be additive.\n\n INPUT:\n\n - ``p`` -- prime >= 5\n\n ... |
def padic_height_pairing_matrix(self, p, prec=20, height=None, check_hypotheses=True):
'\n Computes the cyclotomic `p`-adic height pairing matrix of\n this curve with respect to the basis ``self.gens()`` for the\n Mordell-Weil group for a given odd prime `p` of good ordinary\n reduction.\n The mode... |
def _multiply_point(E, R, P, m):
'\n Computes coordinates of a multiple of `P` with entries in a ring.\n\n INPUT:\n\n - ``E`` -- elliptic curve over `\\QQ` with integer\n coefficients\n\n - ``P`` -- a rational point on `P` that reduces to a\n non-singular point at all primes\n\n - ``R`` -... |
def _multiple_to_make_good_reduction(E):
'\n Return the integer `n_2` such that for all points `P` in `E(\\QQ)`\n `n_2*P` has good reduction at all primes.\n If the model is globally minimal the lcm of the\n Tamagawa numbers will do, otherwise we have to take into\n account the change of the model.... |
def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True):
'\n Compute the cyclotomic `p`-adic height.\n\n The equation of the curve must be integral and minimal at `p`.\n\n INPUT:\n\n - ``p`` -- prime >= 5 for which the curve has\n semi-stable reduction\n\n - ``prec`` -- integer >... |
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