curve #6281

y² = x³ + x² − 20664192704979420841298124661942797x + 1142972628990083443226100921025132463367001234711311

a-invariants: [0, 1, 0, -20664192704979420841298124661942797, 1142972628990083443226100921025132463367001234711311]
c-invariants: [991881249839012200382309983773254272, -987528351447432100898638694799787650642948969430098304]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 163293419186021457311442065268233444059774956
discriminant (Δ): 363837833876940970796652327466105848631057271363958317492930530659553474644209322264976195485295858944
Faltings height: 18.4549
naive height: 248.6547
primes of bad reduction: 2, 3, 7, 13, 17, 23, 31, 41, 47, 73, 197, 359, 389, 757, 15289, 15661, 51631, 26202601
regulator: 548270053118500.32898562470233221507584900159877903813806040854
submitted by: Maksym Voznyy
submitted at: 2026-10-31 23:40:32 UTC
last updated: 2026-10-31 23:40:32 UTC

Witness: 12 independent points log in to add more points →

(141762755028462020277282/674041, -43221038925952571393338671816330453/553387661)
(63297284513897925225617138/718293601, -48366355064067462330952137130334387271/19250986800401)
(20517791865065917030221/198916, -926002151924048681981820294192051/88716536)
(45807117630438879974457/436921, 3278776925061854523384438244876902/288804781)
(31526319270692823474854340091605083265/38467701021150069601, -174457250010737457339913308029963877577336890782809087266/238585665903213922285307383151)
(18708626961632793597161/177241, -876057667934067648610877798452314/74618461)
(3690780603305263462445853/1471369, -7079112262502171882111625574741044114/1784770597)
(2071043844002815888413/14641, 57415214563938663901574680710954/1771561)
(18371133354000003075475370/1247689, -78737805061710541688672862897856184913/1393668613)
(50639914218086934516570106890/435387505921, -5082306772543591238418135931625837282861979/287285656519406719)
(2917571726659298213466/32041, 23185999233521682366251883670569/5735339)
(354548354410405373152995320409/3983309447329, -23656773622583602941391343624850374166165042/7949980611096506767)

Commentary

Voznyy (2026), T = 10728/329 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-10-31 23:41:27 UTC · history