curve #6389

y² = x³ + x² − 603386600145699729x + 139857513590772622141843311

a-invariants: [0, 1, 0, -603386600145699729, 139857513590772622141843311]
c-invariants: [28962556806993587008, -120836891916202886372514142720]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3987245663666297209142592
discriminant (Δ): 5609431884807151068849191111737455056814520759587323904
Faltings height: 9.2175
naive height: 134.4376
primes of bad reduction: 2, 3, 11, 107, 163, 313, 977, 1487, 2017, 10729
regulator: 12049829.407486093690474705952726317372294926913385320572
submitted by: The Wren
submitted at: 2026-11-02 01:10:36 UTC
last updated: 2026-11-02 01:10:36 UTC

Witness: 9 independent points log in to add more points →

(40780060386/25, 7390819898182809/125)
(1848985134642/1681, 1957799794316660085/68921)
(207131613, 4874807288340)
(240419741, 2947561990668)
(1832249749551/625, 2398625618733450576/15625)
(-824927043, 8731596017700)
(-389368137, 17769802541040)
(-493336569, 17817448079808)
(147321380895, 56544796308687984)

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=-121/214. 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 [9,9]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:10:37 UTC · history