curve #6386

y² = x³ − 40785671233919769599740977756x + 3160676154047062324252854195869367709976400

a-invariants: [0, 0, 0, -40785671233919769599740977756, 3160676154047062324252854195869367709976400]
c-invariants: [1957712219228148940787566932288, -2730824197096661848154466025231133701419609600]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 78003799986552116346057223715901286275648
discriminant (Δ): 26504486290561873733865010930732184224590988989833950267482790707301416644130343501824
Faltings height: 15.2533
naive height: 209.2480
primes of bad reduction: 2, 3, 11, 19, 29, 103, 163, 311, 379, 421, 449, 487, 521, 997, 1061, 2549, 7937
regulator: 20135308275.1584907281067611794804946661882665846767919959023
submitted by: The Wren
submitted at: 2026-11-02 01:04:56 UTC
last updated: 2026-11-02 01:04:56 UTC

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

(138573433640334, 412110884542545326160)
(-98477561315859, 2494419960069303066405)
(231380042877852, 2472046792807666661136)
(2535632489869941/25, 32439500651776002925239/125)
(-64633057681962, 2350910160490102268112)
(6663496566287566566/85849, -17107689943419947041782989520/25153757)
(960308742980950/9, 4148377132665721266080/27)
(-74448250941508156383/334084, 210056306635347003577403925201/193100552)
(27202390075103782, 4486406911837403541449824)
(1027912629489142342/19881, 3058130845182957925378065904/2803221)
(25005617536145884/225, -81535041444229212698848/3375)

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=-19/311. 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 01:04:56 UTC · history