curve #6416

y² = x³ − 38577034998339291x + 2645177350652894602644570

a-invariants: [0, 0, 0, -38577034998339291, 2645177350652894602644570]
c-invariants: [1851697679920285968, -2285433230964100936684908480]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 4254224472462433385668752
discriminant (Δ): 651543331035537706902389363677642298194257692550144
Faltings height: 8.4858
naive height: 126.1879
primes of bad reduction: 2, 3, 7, 11, 13, 17, 29, 41, 107, 113, 293, 463, 571, 1559
regulator: 474170.092577881494199105659640816537254090228011078602
submitted by: The Wren
submitted at: 2026-11-02 14:14:29 UTC
last updated: 2026-11-02 14:14:29 UTC

Witness: 8 independent points log in to add more points →

(10304003907, 1045756647681174)
(141788217, 160948008144)
(762446853, 20407395975732)
(-4282970739/25, 256962970609884/125)
(-133898139, 2325929797620)
(3511482955/9, 185079032420590/27)
(648427819, -15819855602950)
(209740660197/961, -64046610753044724/29791)

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=-374/37. 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 [8,8]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 14:14:29 UTC · history