curve #6392

y² = x³ + x² − 3637896793238508241067674137407424x + 59245451577922573057533592890983370830078414726916

a-invariants: [0, 1, 0, -3637896793238508241067674137407424, 59245451577922573057533592890983370830078414726916]
c-invariants: [174619046075448395571248358595556368, -51188070163325104169423300710500005824677901897393600]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2774737613430508387442347675631879178546556272
discriminant (Δ): 1564949344296111487865026593217573450342657457776982755146521439642631825224557564524904033666853356544
Faltings height: 18.3151
naive height: 243.4437
primes of bad reduction: 2, 3, 11, 23, 31, 53, 59, 61, 73, 89, 101, 113, 149, 211, 337, 2099, 2521, 2963, 3469, 39317
regulator: 129942025575.43783111422017406573806076624377609293635247172473
submitted by: The Wren
submitted at: 2026-11-02 01:15:12 UTC
last updated: 2026-11-02 01:15:12 UTC

Witness: 12 independent points log in to add more points →

(12673118384303712, 3895812900074902692361410)
(93768727566871758, 23293625727772273242510240)
(-31014544927079346, 11926450117677234344187840)
(17825085260813462, 251516856959253633966240)
(3751007055139979160, -7263834043943602747961715474)
(-42332569629742992, 11721136450765835758184190)
(14450334434544750, 3113524032254112393843696)
(742398441247180244904/51529, -36661417646867154379740910126554/11697083)
(5296251622133374989/4, 12176077520494416906589231701/8)
(14181005664738105576/1369, -241191367072032849063938011530/50653)
(13939109820998862, 3353320850781389612208480)
(-1609494072421570176/25, 644860851162506251733338182/125)

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=-2024/2963. 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 [12,12]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:15:12 UTC · history