curve #6400

y² = x³ + x² − 950167569313656257737526096x + 11155037997473816446473529613263393162980

a-invariants: [0, 1, 0, -950167569313656257737526096, 11155037997473816446473529613263393162980]
c-invariants: [45608043327055500371401252624, -9637952829817651058013091918861800100330432]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1266010354850622690046273464492857040
discriminant (Δ): 1145176515428088219881658764884671042005727877663279912741610214075126432512640000
Faltings height: 14.3647
naive height: 197.9696
primes of bad reduction: 2, 3, 5, 11, 31, 53, 83, 89, 97, 103, 109, 167, 179, 191, 193, 509, 1723, 3413
regulator: 867416280.650313824853926115178193890653979463941188681252372
submitted by: The Wren
submitted at: 2026-11-02 01:28:13 UTC
last updated: 2026-11-02 01:28:13 UTC

Witness: 11 independent points log in to add more points →

(5551373030122, 77790643163312399100)
(19437415730344, 5472368843282126874)
(-32182451902304, -91662784507661349546)
(579148913615512/121, 109083960587586464773650/1331)
(71265786264110008/289, -18882775990651142266532550/4913)
(11042705570266, 44823873898722563964)
(18736630374129/64, 53397678060541687790313/512)
(26226815556299542/361, 3913722347779502484507600/6859)
(2988563255530255, 163369382427280667886930)
(39357502499872, 186344033046143380650)
(19288829905066, 1999837127260490364)

Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=109/558. Family equations as posted by wgxli with curve #4252; section formulas from the Layr Labs Elliptic Rank Challenge repository (submission/research/x400a-family.gp, commit 7482bf9). T selected by our Mestre-Nagao sieve; PARI/GP 2.19 ellrank seeded with the family sections, bounds [11,11]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:28:13 UTC · history