curve #236

y² = x³ − 40169264421361745502396x + 2062092485931366276599688546201504

a-invariants: [0, 0, 0, -40169264421361745502396, 2062092485931366276599688546201504]
rank (lower bound): ≥ 16
conductor (N): 2329523824211488655524107381802033064249929281525454944
naive height: 167.7558
Faltings height: 12.0132
discriminant (Δ): 2311256994351959597357327359870479823158106106819333943959339451297792
primes of bad reduction: 2, 3, 11, 19, 31, 37, 79, 163, 1187, 2207494092079386051158169308866468641719
regulator: 1575306794443644.152922817277863822046611524738222651298257620999230147
submitted by: Seewoo Lee
last updated: 2026-07-01 22:38:40

Witness: 16 independent points

(263473897774, 98835728473299268)
(1694406292782, 2190584624758308540)
(107788063883300184/16129, 35372203232200059004682544/2048383)
(177077545644550110/1026169, 17045499838668802241704524/1039509197)
(698820360270, -561480046717032828)
(6318338595346081/8100, -486498513485524345029521/729000)
(5853814809142294/34969, 35076549330988415699084/6539203)
(316593635550, -145180658240849052)
(12138887762392620/63001, 607416600498386153991228/15813251)
(182224971549, -28163823213797307)
(25789670918080633836/5058001, -130868856464406362882549387748/11375444249)
(63166954843761178486/375623161, -57635319338620210678550548852/7279952483341)
(431450722493976/961, -8132585828692923037728/29791)
(667137117769/4, -11108990118817691/8)
(260509642146528246/1428025, -48423597185946377910988044/1706489875)
(1244240230476587934/3545689, -1178907764805735000176562444/6676532387)

Commentary

Rank ≥16. 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,-4), cleared tuple [-210, -120, 330, 875, -864, -11], shift t=606/1. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 16 via 16 independent points from a quartic point search + Néron–Tate height-pairing matrix.

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