Elliptic curves

• Elliptic curve constructor
  • EllipticCurveFactory
  • EllipticCurve_from_Weierstrass_polynomial()
  • EllipticCurve_from_c4c6()
  • EllipticCurve_from_cubic()
  • EllipticCurve_from_j()
  • EllipticCurves_with_good_reduction_outside_S()
  • are_projectively_equivalent()
  • chord_and_tangent()
  • coefficients_from_Weierstrass_polynomial()
  • coefficients_from_j()
  • projective_point()
  • tangent_at_smooth_point()
• Construct elliptic curves as Jacobians
  • Jacobian()
  • Jacobian_of_curve()
  • Jacobian_of_equation()
• Points on elliptic curves
  • EllipticCurvePoint
  • EllipticCurvePoint_field
  • EllipticCurvePoint_finite_field
  • EllipticCurvePoint_number_field
• Elliptic curves over a general ring
  • EllipticCurve_generic
  • is_EllipticCurve()
• Elliptic curves over a general field
  • EllipticCurve_field
  • compute_model()
• Elliptic curves over finite fields
  • EllipticCurve_finite_field
  • fill_ss_j_dict()
  • is_j_supersingular()
  • supersingular_j_polynomial()
• Formal groups of elliptic curves
  • EllipticCurveFormalGroup

Maps between them

• Elliptic-curve morphisms
  • EllipticCurveHom
  • compare_via_evaluation()
• Isomorphisms between Weierstrass models of elliptic curves
  • WeierstrassIsomorphism
  • baseWI
  • isomorphisms()
• Isogenies
  • EllipticCurveIsogeny
  • compute_codomain_formula()
  • compute_codomain_kohel()
  • compute_intermediate_curves()
  • compute_isogeny_kernel_polynomial()
  • compute_isogeny_starks()
  • compute_sequence_of_maps()
  • compute_vw_kohel_even_deg1()
  • compute_vw_kohel_even_deg3()
  • compute_vw_kohel_odd()
  • fill_isogeny_matrix()
  • isogeny_codomain_from_kernel()
  • split_kernel_polynomial()
  • two_torsion_part()
  • unfill_isogeny_matrix()
• √élu algorithm for elliptic-curve isogenies
  • EllipticCurveHom_velusqrt
  • FastEllipticPolynomial
  • ProductTree
  • prod_with_derivative()
• Composite morphisms of elliptic curves
  • EllipticCurveHom_composite
• Isogenies of small prime degree
  • Fricke_module()
  • Fricke_polynomial()
  • Psi()
  • Psi2()
  • is_kernel_polynomial()
  • isogenies_13_0()
  • isogenies_13_1728()
  • isogenies_2()
  • isogenies_3()
  • isogenies_5_0()
  • isogenies_5_1728()
  • isogenies_7_0()
  • isogenies_7_1728()
  • isogenies_prime_degree()
  • isogenies_prime_degree_general()
  • isogenies_prime_degree_genus_0()
  • isogenies_prime_degree_genus_plus_0()
  • isogenies_prime_degree_genus_plus_0_j0()
  • isogenies_prime_degree_genus_plus_0_j1728()
  • isogenies_sporadic_Q()

Elliptic curves over number fields

• Elliptic curves over the rational numbers
  • EllipticCurve_rational_field
  • cremona_curves()
  • cremona_optimal_curves()
  • elliptic_curve_congruence_graph()
  • integral_points_with_bounded_mw_coeffs()
• Tables of elliptic curves of given rank
  • EllipticCurves
• Elliptic curves over number fields
  • EllipticCurve_number_field
• Canonical heights for elliptic curves over number fields
  • EllipticCurveCanonicalHeight
  • UnionOfIntervals
  • eps()
  • inf_max_abs()
  • min_on_disk()
  • nonneg_region()
  • rat_term_CIF()
• Saturation of Mordell-Weil groups of elliptic curves over number fields
  • EllipticCurveSaturator
  • p_projections()
  • reduce_mod_q()
• Torsion subgroups of elliptic curves over number fields (including \(\QQ\))
  • EllipticCurveTorsionSubgroup
  • torsion_bound()
• Galois representations attached to elliptic curves
  • GaloisRepresentation
• Galois representations for elliptic curves over number fields
  • Billerey_B_bound()
  • Billerey_B_l()
  • Billerey_P_l()
  • Billerey_R_bound()
  • Billerey_R_q()
  • Frobenius_filter()
  • GaloisRepresentation
  • deg_one_primes_iter()
  • reducible_primes_Billerey()
  • reducible_primes_naive()
• Isogeny class of elliptic curves over number fields
  • IsogenyClass_EC
  • IsogenyClass_EC_NumberField
  • IsogenyClass_EC_Rational
  • isogeny_degrees_cm()
  • possible_isogeny_degrees()
• Tate-Shafarevich group
  • Sha
• Complex multiplication for elliptic curves
  • cm_j_invariants()
  • cm_j_invariants_and_orders()
  • cm_orders()
  • discriminants_with_bounded_class_number()
  • hilbert_class_polynomial()
  • is_cm_j_invariant()
  • largest_fundamental_disc_with_class_number()
• Testing whether elliptic curves over number fields are \(\QQ\)-curves
  • Step4Test()
  • conjugacy_test()
  • is_Q_curve()

The following relate to elliptic curves over local nonarchimedean fields.

• Local data for elliptic curves over number fields
  • EllipticCurveLocalData
  • check_prime()
• Kodaira symbols
  • KodairaSymbol()
  • KodairaSymbol_class
• Tate’s parametrisation of \(p\)-adic curves with multiplicative reduction
  • TateCurve

Analytic properties over \(\CC\).

• Weierstrass \(\wp\)-function for elliptic curves
  • compute_wp_fast()
  • compute_wp_pari()
  • compute_wp_quadratic()
  • solve_linear_differential_system()
  • weierstrass_p()
• Period lattices of elliptic curves and related functions
  • PeriodLattice
  • PeriodLattice_ell
  • extended_agm_iteration()
  • normalise_periods()
  • reduce_tau()
• Regions in fundamental domains of period lattices
  • PeriodicRegion

Modularity and \(L\)-series over \(\QQ\).

• Modular parametrization of elliptic curves over \(\QQ\)
  • ModularParameterization
• Modular symbols attached to elliptic curves over \(\QQ\)
  • ModularSymbol
  • ModularSymbolECLIB
  • ModularSymbolSage
  • modular_symbol_space()
• Modular symbols by numerical integration
  • ModularSymbolNumerical
• \(L\)-series for elliptic curves
  • Lseries_ell
• Heegner points on elliptic curves over the rational numbers
  • GaloisAutomorphism
  • GaloisAutomorphismComplexConjugation
  • GaloisAutomorphismQuadraticForm
  • GaloisGroup
  • HeegnerPoint
  • HeegnerPointOnEllipticCurve
  • HeegnerPointOnX0N
  • HeegnerPoints
  • HeegnerPoints_level
  • HeegnerPoints_level_disc
  • HeegnerPoints_level_disc_cond
  • HeegnerQuatAlg
  • HeegnerQuatAlgEmbedding
  • KolyvaginCohomologyClass
  • KolyvaginCohomologyClassEn
  • KolyvaginPoint
  • RingClassField
  • class_number()
  • ell_heegner_discriminants()
  • ell_heegner_discriminants_list()
  • ell_heegner_point()
  • heegner_index()
  • heegner_index_bound()
  • heegner_point()
  • heegner_point_height()
  • heegner_points()
  • heegner_sha_an()
  • is_inert()
  • is_kolyvagin_conductor()
  • is_ramified()
  • is_split()
  • kolyvagin_point()
  • kolyvagin_reduction_data()
  • make_monic()
  • nearby_rational_poly()
  • quadratic_order()
  • satisfies_heegner_hypothesis()
  • satisfies_weak_heegner_hypothesis()
  • simplest_rational_poly()
• \(p\)-adic \(L\)-functions of elliptic curves
  • pAdicLseries
  • pAdicLseriesOrdinary
  • pAdicLseriesSupersingular

To be sorted

• Descent on elliptic curves over \(\QQ\) with a 2-isogeny
  • test_els()
  • test_padic_square()
  • test_qpls()
  • test_valuation()
  • two_descent_by_two_isogeny()
  • two_descent_by_two_isogeny_work()
• Elliptic curves with prescribed good reduction
  • curve_key()
  • egros_from_j()
  • egros_from_j_0()
  • egros_from_j_1728()
  • egros_from_jlist()
  • egros_get_j()
  • is_possible_j()
• Elliptic curves over padic fields
  • EllipticCurve_padic_field
• Denis Simon’s PARI scripts
  • init()
  • simon_two_descent()
• Elliptic curves with congruent mod-5 representation
  • mod5family()
• Morphism to bring a genus-one curve into Weierstrass form
  • WeierstrassTransformation
  • WeierstrassTransformationWithInverse()
  • WeierstrassTransformationWithInverse_class

Hyperelliptic curves

• Hyperelliptic curve constructor
  • HyperellipticCurve()
• Hyperelliptic curves over a general ring
  • HyperellipticCurve_generic
  • is_HyperellipticCurve()
• Hyperelliptic curves over a finite field
  • HyperellipticCurve_finite_field
• Hyperelliptic curves over a \(p\)-adic field
  • HyperellipticCurve_padic_field
• Hyperelliptic curves over the rationals
  • HyperellipticCurve_rational_field
• Mestre’s algorithm
  • HyperellipticCurve_from_invariants()
  • Mestre_conic()
• Computation of Frobenius matrix on Monsky-Washnitzer cohomology
  • MonskyWashnitzerDifferential
  • MonskyWashnitzerDifferentialRing
  • MonskyWashnitzerDifferentialRing_class
  • SpecialCubicQuotientRing
  • SpecialCubicQuotientRingElement
  • SpecialHyperellipticQuotientElement
  • SpecialHyperellipticQuotientRing
  • SpecialHyperellipticQuotientRing_class
  • adjusted_prec()
  • frobenius_expansion_by_newton()
  • frobenius_expansion_by_series()
  • helper_matrix()
  • lift()
  • matrix_of_frobenius()
  • matrix_of_frobenius_hyperelliptic()
  • reduce_all()
  • reduce_negative()
  • reduce_positive()
  • reduce_zero()
  • transpose_list()
• Frobenius on Monsky-Washnitzer cohomology of a hyperelliptic curve over a
  • hypellfrob()
  • interval_products()
• Jacobian of a general hyperelliptic curve
  • HyperellipticJacobian_generic
• Jacobian of a hyperelliptic curve of genus 2
  • HyperellipticJacobian_g2
• Rational point sets on a Jacobian
  • JacobianHomset_divisor_classes
• Jacobian ‘morphism’ as a class in the Picard group
  • JacobianMorphism_divisor_class_field
  • cantor_composition()
  • cantor_composition_simple()
  • cantor_reduction()
  • cantor_reduction_simple()
• Hyperelliptic curves of genus 2 over a general ring
  • HyperellipticCurve_g2
• Compute invariants of quintics and sextics via ‘Ueberschiebung’
  • Ueberschiebung()
  • absolute_igusa_invariants_kohel()
  • absolute_igusa_invariants_wamelen()
  • clebsch_invariants()
  • clebsch_to_igusa()
  • differential_operator()
  • diffsymb()
  • diffxy()
  • igusa_clebsch_invariants()
  • igusa_to_clebsch()
  • ubs()
• Kummer surfaces over a general ring
  • KummerSurface
• Conductor and reduction types for genus 2 curves
  • Genus2reduction
  • ReductionData
  • divisors_to_string()

Indices and Tables

• Index

• Module Index

• Search Page