curve #6393

y² = x³ + x² − 8641853357962512066041822056x + 280467632210410873874388895227789394920644

a-invariants: [0, 1, 0, -8641853357962512066041822056, 280467632210410873874388895227789394920644]
c-invariants: [414808961182200579170007458704, -242324034229797483881239098680285057256188608]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1886403763909308393938467715492012880
discriminant (Δ): 7322788127167554754078944674892130344488886932342036131835392444688498190568525440000
Faltings height: 15.0196
naive height: 204.5928
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 59, 61, 83, 101, 107, 127, 131, 149, 227, 233, 503, 14249
regulator: 490404616.903962885568192129131608510926784254950646912483104
submitted by: The Wren
submitted at: 2026-11-02 01:15:50 UTC
last updated: 2026-11-02 01:15:50 UTC

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

(16574362207454, 376546888153394966700)
(-72563592983446, 724892940509600200800)
(72742171967054, 191703138521656307700)
(85864365665822, 414115520949751778388)
(-102965845017346, 280433692579682371500)
(27070021034954, -257621842076141650800)
(22640304285341/4, -3851112368044038064275/8)
(835617302793886/9, 14248230513870690914500/27)
(69113552761304, 115464231945780460050)
(136906479760304, 1289741315160914932050)
(61361937009737281730/97969, 475625325764955831009748160496/30664297)

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=1016/529. 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 01:15:50 UTC · history