curve #90

y² + xy + y = x³ − x² − 1049634928262201481608261153x + 12127108227154239133795359409449990039537

a-invariants: [1, -1, 1, -1049634928262201481608261153, 12127108227154239133795359409449990039537]
rank (lower bound): ≥ 19
conductor (N): 1751718228241340476861580569821746434717712624403433485131514999870
naive height: 198.2683
Faltings height: 14.4808
discriminant (Δ): 10477910772405735373306877059383665173459657501610235009100067004657607156983398400
primes of bad reduction: 2, 3, 5, 19, 37, 8574497, 181208356857259, 4413293403451386737, 4037539076961879275231
regulator: 10371257585479461054.35598398694932580359550898384903385812428978701472381867
submitted by: David Renshaw
last updated: 2026-07-01 22:30:42

Witness: 19 independent points

(-8779463831497, 143755449342791785188)
(24240245774135, 30446890111192735716)
(-22245461900617, 156423538644219166308)
(36555310855883, -150352607095203320592)
(-8080699457113, 141708301267997242020)
(-14846294273337, 156326553803804543268)
(36476994847767, 149582124592754048612)
(-34924411096447, 78658644953248301238)
(59607687170287/9, -1996141721696772632564/27)
(-26267158673455, 146883134785122482568)
(-16834995329497, -158197280846105092812)
(342081548339663, 6299474080859475015468)
(14189440889847, 9501613201130832932)
(-20800327426857, -157988981093285571132)
(526338098596415, 12052885365530882707908)
(9835031446475, -52490357206977923004)
(-37100686104889, -1280640675347457756)
(-36314897406925, -48512158639169211804)
(109946355750407/49, 33926203667374622610684/343)

Commentary

New curve of rank >= 19 and naive height 198.268, found as the specialization T = 3251/8 (t = 3251/16) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by an exhaustive minimal-height scan crossed with a staged Nagao-sum sieve; 19 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 18:51:20 · history