curve #3639

y² + xy + y = x³ + x² − 419672873567864790074815x + 104614921796713109015272026649125797

a-invariants: [1, 1, 1, -419672873567864790074815, 104614921796713109015272026649125797]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 1284522297763555835148164816047706661990
discriminant (Δ): 2631482848787910831318696140654317366291881968892199437514982400000000
Faltings height: 12.2950
naive height: 174.7949
primes of bad reduction: 2, 3, 5, 11, 31, 59, 73, 191, 193, 487, 641, 691, 751, 4951, 15161, 340519
regulator: 18488679873557.02769566042068264331069808071407486770963879712768
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:24:50 UTC
last updated: 2026-09-29 19:24:50 UTC

Witness: 13 independent points log in to add more points →

(-739980079813, 99866341560850626)
(-695453973793, 245188933397930646)
(-418062552343, 454969449064279746)
(1447210593203/4, 93482152657207293/8)
(363278237755, 9955746927195138)
(379103896907, 150947988673746)
(510021524027, 152448159655536066)
(3224138878843/9, 423856368128490862/27)
(391552054771267/81, 7681631728225505255434/729)
(2488609601429275/6241, 12784130083356131744862/493039)
(3394068702935107/10201, 43886305603604409781946/1030301)
(7481805254580955/22201, 126295256399581434059762/3307949)
(15899482391642627/44521, 158979692904498319859766/9393931)

Commentary

Specialization at T = -23/90 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 19:24:50 UTC · history