curve #206

y² + xy = x³ − 374081627322112870915013649086x + 84694590726964421851283320326657184990555716

a-invariants: [1, 0, 0, -374081627322112870915013649086, 84694590726964421851283320326657184990555716]
rank (lower bound): ≥ 12
conductor (N): 431184740183886106535857306115602438132236071442111719243984779367694673530
naive height: 215.8964
Faltings height: 15.9179
discriminant (Δ): 251453575815775763875321183856781238190717796098320943436148000167676873962388406215526400
primes of bad reduction: 2, 3, 5, 7, 19, 31, 157, 4517, 21447358407501679739, 229195053659697773915482188781052624679234107
regulator: 128581143130572.87247198242309993116966404006654526604274397426
submitted by: Seewoo Lee
last updated: 2026-07-01 22:47:36

Witness: 12 independent points

(1311962006756715124/9, 1502720013154061565338633534/27)
(1622964775848764290/81, 2066633657734054159603682216/729)
(2072780699018820625/576, 2943546468634289028128158229/13824)
(2186299717178498316/841, 3149957659531808431157396778/24389)
(21543146691997293/16, 2877874286932028601295563/64)
(25660979318382673047964/62520649, -270616637356825393764260426373754/494350771643)
(29542255198291148257780/69839449, -854884756696283163043353677345782/583648275293)
(36612916576974039331564/79798489, -2214006161797201672783134021818326/712839902237)
(539247600876988, -6307035117458519888078)
(561618735181336, -7193514911777232202958)
(29051022192219566367268/57623281, 2153653151864037845479171859549542/437418326071)
(3094088113068047834300476/664041361, 5397713329122196810957917667676278178/17111681831609)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=7907/3: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:10:53 · history