curve #239

y² = x³ − x² − 22359392011092389449x + 40692284642676185167439218585

a-invariants: [0, -1, 0, -22359392011092389449, 40692284642676185167439218585]
rank (lower bound): ≥ 15
conductor (N): 493331968419716917139720664887174469134762256768
naive height: 145.2749
Faltings height: 9.7861
discriminant (Δ): 85738128226224787181909938997870961494015365496541683712
primes of bad reduction: 2, 3, 7, 13, 29, 4868091091, 13751713379, 7271987713848853806487
regulator: 72348684025139.813690956329432821964132838122276072356655182051278552
submitted by: Seewoo Lee
last updated: 2026-07-01 22:42:37

Witness: 15 independent points

(10851060085/4, -258818176155/8)
(7558450841269/289, 20461735427020481412/4913)
(36592676846697/1024, -219509068333802137163/32768)
(12690550357/4, -328743630743307/8)
(-21288072143/16, -16699737845264319/64)
(1634480901, -92265088908044)
(-2111327789, -280158194286516)
(19227744004225597/3261636, -1986693395508881051828039/5890514616)
(40409079012536/19321, -148973071535977430301/2685619)
(208147940321649/76729, 390225018276560012/21253933)
(-134774415986415/139129, -12863588512365650166380/51895117)
(1921166487515869/822649, -25976048739534448334052/746142643)
(269290852319401/103041, -342687181344661622924/33076161)
(60150234401830765/21353641, -764007110263969721150820/98675175061)
(4103100327, -134264379870296)

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)=(1,-3), cleared tuple [20, -60, 40, 125, 3, -128], shift t=213. Two depressed cubics with equal product (satisfies Mestre 2e5=e2e3 identically). Certified rank 15 via 15 independent points (quartic point search + Néron–Tate height-pairing matrix). Smallest-conductor rank-15 of the family.

last edited by Seewoo Lee at 2026-06-26 15:03:44 · history