curve #445

y² + xy = x³ − 90872683182171123779863585495688745x + 10534068423995936974583664932054639017801516408581081

a-invariants: [1, 0, 0, -90872683182171123779863585495688745, 10534068423995936974583664932054639017801516408581081]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 3038587437160140161378537018221207898070176008185832826819943447344543898
discriminant (Δ): 88829305886035909223406790580188831734489544997532961921266764802355273349240637700988946188606271946752
Faltings height: 18.8621
naive height: 253.0979
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 47, 103, 15648087912489970555684237500289775534019998199017993121553
regulator: 7204154760760603178.67429124959362750014293091953483704426642506491967324350154
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:54:59 UTC
last updated: 2026-08-30 15:54:59 UTC

Witness: 20 independent points log in to add more points →

(168995182839532794, -1849047815228328206705085)
(193685140022916858, -14116356344573555684028861)
(286764959620059210, 89759888056250214764740179)
(-290566955614904094, 111383771805579971481344403)
(168867936408690186, 2021923402283802959216403)
(-348049312491962766, -364651622937313662507429)
(675843549869225715/4, -15174133203965767459143693/8)
(226743412114140666, 39834015980031018816673731)
(-345228833338297164, -27578051745191924150290989)
(264494672967316115418/1681, 789406769436700456094799249579/68921)
(117088567945331298, 38719104511344866214092043)
(153533084142806970, -14186246948927756870278269)
(4705528746139000386/25, 1237790742135157721822446119/125)
(283719461313430458, 87121807406646082412817603)
(156725078089691856, 11901217671025861648745403)
(187535709998017620, -9367985213511861833743719)
(169343824650362106, 1292130346674654571447491)
(677728207348789695/4, -8931655065396010855482123/8)
(-339654662569037526, -47064215388506264365852125)
(390844227572272218, -186338563793494941720656637)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 22/5. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:54:59 UTC · history