curve #238

y² + xy + y = x³ − x² − 25674766841623769204676x + 1584473203460742150597373002163815

a-invariants: [1, -1, 1, -25674766841623769204676, 1584473203460742150597373002163815]
c-invariants: [1232388808397940921824433, -1368984847784535468478339539721326441]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 95954989391736297575743447520151210094560387026536946
discriminant (Δ): -1382730624930385367184147512181586170312381345872576399281800347648
Faltings height: 11.6251
naive height: 166.4143
primes of bad reduction: 2, 7, 13, 23, 67, 614071, 2845615572511, 195793702136100389221771138343
regulator: 91113336322005.277313711589725666999350825220758797042718744371374129
submitted by: Seewoo Lee
submitted at: 2026-06-26 06:57:35 UTC
last updated: 2026-07-01 22:42:38 UTC

Witness: 15 independent points log in to add more points →

(4686108375477/49, -664183300877107717/343)
(131018426453, 21671238900850445)
(721984601739957/5329, -9471872484708889421835/389017)
(2694575359197861/2209, -138723138874027700845101/103823)
(190686676611340509/2247001, -13804565079891464631408393/3368254499)
(14987765522285/169, -4920895085425106975/2197)
(24227790657649/225, -27903398879047177237/3375)
(3819910387281869/42025, -11330999487134645156863/8615125)
(77611481861, -7701501794569091)
(2638222669560109/26569, -16264401587478186117297/4330747)
(271036990440985/289, -4400877261188912276575/4913)
(229833258657625885/2627641, -12003197530901784628769351/4259406061)
(16035175610211357/35344, 1918001943185220915702165/6644672)
(1552119729577773/17161, -3324258507528512596305/2248091)
(121581860292421693/961, -42393848878672419779626805/29791)

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)=(-7,-1), cleared tuple [-210, -30, 240, 189, -125, -64], shift t=1791/4. 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 06:57:35 UTC · history