curve #96

y² + xy + y = x³ − 3764690650150480844855x + 89019354183527481401653208626710

a-invariants: [1, 0, 1, -3764690650150480844855, 89019354183527481401653208626710]
c-invariants: [180705151207223080553017, -76912722014838801657839206874307181]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 25777625822377588725485551545783271350685691069629753526342
discriminant (Δ): -8540075879702050389575912256014906231939468079986198083770051916
Faltings height: 11.1696
naive height: 160.6560
primes of bad reduction: 2, 11, 37, 107, 12412556554267617257, 23843691413839155824600324843196647
regulator: 19359556026346100.6303303098718257240779383918436146546525416442360173901
submitted by: David Renshaw
submitted at: 2026-06-23 13:32:21 UTC
last updated: 2026-07-01 22:34:42 UTC

Witness: 17 independent points log in to add more points →

(90465122187, -22109028378030402)
(29636398430, 1864819852756660)
(26054573840, 2935775090470274)
(225363887573, 103375773706825504)
(238623691977, -113040864973820416)
(17360593283, -5375360169896206)
(14618830545, -6091655100390690)
(-70533645811, -1911027879550192)
(126855903943/4, 9873344264724949/8)
(164303427527/4, -15358735695850363/8)
(33812349577, 618959095388288)
(37672630837, 812121700571498)
(624752627246/25, -405930157653036359/125)
(33831546370, -613808681238931)
(103707900164, -28530802496948098)
(27298730108, -2567436092943578)
(11197315494189/625, 81673509203033696688/15625)

Commentary

Generated by local generalized Mestre/Fermigier quartic-family search: Mestre parameters u=-2, v=-4, specialization T=1142. Found in uv-smoke-5 by Nagao stage-1 sieve (M=200, m<=5000, n<=12), rescored at M=1000, selected by exact minimal-height triage, and certified with famcert from quartic-side rational points.

last edited by David Renshaw at 2026-06-23 13:32:21 UTC · history