curve #194

y² = x³ − x² − 11524988157418506896x + 15103590903163497035477292096

a-invariants: [0, -1, 0, -11524988157418506896, 15103590903163497035477292096]
rank (lower bound): ≥ 15
conductor (N): 10190033470666423893407916147715865053798375920
naive height: 143.2926
Faltings height: 9.7552
discriminant (Δ): -575294894722450744401258092356189386715188315557054668800
primes of bad reduction: 2, 3, 5, 13, 17, 29, 31, 37, 1213, 228054274471183, 20879094585820301449
regulator: 14815863059320.462760657204460902116663737179414860667111264912466988
submitted by: Seewoo Lee
last updated: 2026-07-01 22:42:37

Witness: 15 independent points

(678134618389, -558429614392973290)
(773401598709/4, -680050397559446355/8)
(12239211117264/529, -42381308250908588520/12167)
(12544641506, -1358179764705030)
(3569890425248/361, -6390921341055807816/6859)
(5591956987984/729, -12102145909255225720/19683)
(2898632381921/1681, 1303373127022665840/68921)
(548698001744/289, 39940953579607080/4913)
(79509693083361/38416, 81170157156022503465/7529536)
(62974323862784/29929, 67360935645521657880/5177717)
(124565294906581/50625, 461664469936998565354/11390625)
(194522633305405/58564, 1644752730648824672287/14172488)
(5865379264, 386380504805240)
(3564819679739280/1907161, 25251714986265769716936/2633789341)
(157879158605/121, 63656031680098026/1331)

Commentary

Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2352/1: 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 15 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 16:09:55 · history