Revision #129 → #1180 · back to history
addedElliptic curve (genus one with point O)29ba73cfa5b7
addedNon-singularity conditionb3d142c4f8ee
addedElliptic curve is an abelian varietyac63ba9d4b5e
addedy² = P(x), deg 3, distinct roots gives elliptic curve3b23ff6ba75a
addedComplex elliptic curves correspond to torus embeddings3c17faa11c64
addedReal elliptic curves as ellipses in hyperbolic plane24a6641b513f
addedTopology: elliptic curve is a toruseeaf76ca7fb8
addedWeierstrass normal forma1cc1ed183db
addedNon-singular iff discriminant nonzeroaf5f50cd5c9f
addedNumber of real components from discriminant sign1120c88bf08a
addedDiscriminants 64 and −368904bd64a6904
addedPoint at infinity is unique intersection with line at infinitya6ffa7196347
addedPoint at infinity is group identity3254d0db224b
addedInverse −P as opposite pointf7274666ea74
addedAddition P + Q via chord8d54f747d122
addedSpecial cases of the addition lawae056e251e01
addedK-rational pointse17a1fd031c0
addedE(K) is a group9df7d1bf004c
addedE(K) is a subgroup of E(L)6b24e205cd6c
addedAddition formula, distinct x (case 1)9b695884c56b
addedInverse by reflection across x-axis (cases 3,4)72d3c44ef961
addedDoubling formula via tangent (case 2)8ca331b8417f
addedGeneral slope expression for cases 1 and 2811fe669238a
addedAddition formulas in characteristic 358cffdb6f05c
addedGroup law on general cubic via inflection pointb6d42958a9e2
addedGroup law on Hessian curves781b81c7ee1b
addedRational points form a subgroup of real points36d743e6eba7
addedy² = x³ + 17 integral solutions411a3e6c828e
addedLjunggren's equation solutions783f60b07348
addedMordell–Weil theoremc5847ddeaf91
addedStructure via fundamental theorem of fg abelian groupsf854bca0961e
addedWeak Mordell–Weil theorem9c54c4ad4262
addedFiniteness of bounded-height rational pointsf7933268e0d2
addedRank of E(Q)058f829a558c
addedCurve of rank 206c55ade4d20f
addedMazur's torsion theoremc9efbb796c44
addedBirch and Swinnerton-Dyer conjectured8dd8b5b818a
addedHasse–Weil zeta function2de528b97df9
addedBad reduction at pe63146a33996
addedZeta function over a finite fieldf95dcd1a1f83
addedZeta function is rational in T3261604eae6d
addedTrace of Frobenius308cc26fb7b0
addedL-function of E over Q4796cfabbcc7
addedConductor of E44a34d5b021c
addedBad reduction at 1700a1568879c8
addedHasse's conjecture (analytic continuation)2d56f716a2d9
addedShimura–Taniyama–Weil (modularity)1624ddff9a9b
addedBSD: vanishing order equals rank6f5a3e3462d4
addedCongruent numbercfb85bec938e
addedCongruent number iff rational point of infinite order123582ad3eff
addedTunnell's theorem29834774f3ef
addedAverage rank under GRH and BSD3728468bf962
addedHasse's theorem on elliptic curvesb555cefd894a
addedE(F_q) is cyclic or product of two cyclic groups3750d58d8355
addedCurve over F71 with 72 points4dcfab2d54c2
addedLocal zeta function of E over F_q833325c35dae
addedLocal zeta function is rational in T3bc5b628b7d4
addedZeta function of y²+y=x³ over F28b0eec21dd32
addedFunctional equation of zeta functiona19a7edd51ec
addedSato–Tate conjecture8f3ca98479cf
addedElliptic curve over a general field0d6cbd0aba86
addedNormal form when char ≠ 2,37d9309228ece
addedGeneral equation in characteristic 37e2c6e87ba3f
addedGeneral equation in characteristic 2735366ef7673
addedK-rational points (general field)f44190eebf0c
addedMordell–Weil over a number field071de695f9b0
addedMerel's theorem on torsion6f566f6864a2
addedSiegel's theorem on integral pointsd2e29cc267e3
addedWeierstrass ℘ differential relationa328950c1ebd
addedWeierstrass functions doubly periodic on torus223f42603d8e
addedEmbedding map is group isomorphism9207bbf9a3aa
addedScaled lattices give isomorphic curves85e3598e6233
addedIsomorphism classes by j-invariantf2c0ba6fc8b4
addedModular invariants g2, g35e87b50e601c
addedj(2i) = 287496c591e8a9d188
addedModular discriminant is transcendental943c7501cb77
addedUniformization theorem for genus-one surfaces6771ba13ffcb
addedn-torsion points on the lattice torus82c961c7d477
addedFundamental periods via arithmetic–geometric mean5887b1dda4bb
addedNine inflection points and Hesse configuration0a75fec11ee3
addedDual isogeny5aaf0c0a60c4
addedExplicit construction of dual isogenyc4bbabd4d749
addedPicard map descends to an isomorphism56f29de2dd69
addedConjugate relation of dual isogeny20e5370692fd