curve #237

y² = x³ + x² − 1651640332674597876x + 910715925208661721534017824

a-invariants: [0, 1, 0, -1651640332674597876, 910715925208661721534017824]
rank (lower bound): ≥ 15
conductor (N): 5919531324154970229554784141330370952904285996920
naive height: 137.6757
Faltings height: 9.4469
discriminant (Δ): -69948023501250722618129515710347501757574437396885241600
primes of bad reduction: 2, 5, 13, 19, 23, 26049688981495204319463052901471444080726483
regulator: 111257299159024.59816588442634390170741817967774122122164844181174948
submitted by: Seewoo Lee
last updated: 2026-07-01 22:41:39

Witness: 15 independent points

(772668450, 9789861575182)
(682982520231464/143641, -17263025154188599268920/54439939)
(2306049782, 96774035268922)
(150596914681/64, 51336000912478035/512)
(681402367466/49, 560166370225180550/343)
(2852440687, 139313169639822)
(18862882604/25, 1212361838592208/125)
(5397415438786/23409, 83394215197427708990/3581577)
(78607545570, 22036301809608398)
(28772514445/36, 2172923165525957/216)
(-87485153086/841, 802024535920980970/24389)
(17438427206/25, 1237612245267346/125)
(1043724563974/441, 937999906284868750/9261)
(16728707607069638/165649, 2163508515105097434676454/67419143)
(390658744, 18030759283680)

Commentary

Rank ≥15. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-10,-1), cleared tuple [-840, -84, 924, 640, -343, -297], shift t=784/1. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 15 via 15 independent points from a quartic point search + Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-26 06:57:31 · history