curve #208

y² + xy = x³ − 304695382156188407658584613236x + 64760983700857400691050362439563148137462416

a-invariants: [1, 0, 0, -304695382156188407658584613236, 64760983700857400691050362439563148137462416]
rank (lower bound): ≥ 12
conductor (N): 1107988077126197589769596508555487707559725219449598542068453069615896859730
naive height: 215.2817
Faltings height: 15.6797
discriminant (Δ): -1388810570434782811007607331048073936871115674258540673227450623892810201037842324019200
primes of bad reduction: 2, 3, 5, 23, 31, 157, 4513, 11249219, 1500692293891444032300619619, 4330572287574020219420543749507
regulator: 147652040099442.05947295954239201960603808166881615690156257635
submitted by: Seewoo Lee
last updated: 2026-07-01 22:47:35

Witness: 12 independent points

(44278949867297760957064/935089, -9316796137756519793643768214532204/904231063)
(158955475956867724/9, -63343723260583635566586476/27)
(7690985079964216766320/5368489, -630406900594136415745523463507692/12438789013)
(77834314090789836/361, 20678751759625129232805588/6859)
(299913011917936, 596149116860699711812)
(53714858859344400/169, 350978976489095631299388/2197)
(47471468885246317/144, 649332284048410916301671/1728)
(399524613364744, 2607635597175266655628)
(656000246676580, 12131836808507316807904)
(15043833852323541/16, 1580027181914339609144013/64)
(101292582467515835873416/80263681, 29562434295869282672212131124865908/719082318079)
(1139227591149379/4, 8328224915694189545111/8)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=967/9: 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 16:11:03 · history