curve #3626

y² + xy = x³ + x² − 10890191097078268972x + 13664254157182253537901636340

a-invariants: [1, 1, 0, -10890191097078268972, 13664254157182253537901636340]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 1758821889262453385218180898874
discriminant (Δ): 1998785656382426536316185289523196368038430303785897912172
Faltings height: 9.8002
naive height: 143.1168
primes of bad reduction: 2, 3, 7, 11, 23, 43, 67, 83, 101, 193, 317, 353, 977, 16349, 19867
regulator: 123555300634.745014009534653514819991954703916210495836392039
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:22:05 UTC
last updated: 2026-09-29 19:22:05 UTC

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

(-3464888264, 98996703655114)
(-2136223080, 164863394160730)
(1454569048, 30020172774010)
(194958535704, 86069853692658254)
(-4415185658936/2809, 24416960399789032298/148877)
(19035807246/25, 9530801636606804/125)
(805350423047/484, 130643297899744265/10648)
(1981108094520/1681, 3423009335010014030/68921)
(5605289921443/3249, 618664844481406841/185193)
(104148391583314/36481, 532570967508362362292/6967871)
(755063997469576/2809, 20746395243403910625614/148877)

Commentary

Specialization at T = -62/9 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:22:05 UTC · history