curve #462

y² + xy = x³ − 5831269123044942826740026401456095x + 133924295711655587194437025768409363893193455880025

a-invariants: [1, 0, 0, -5831269123044942826740026401456095, 133924295711655587194437025768409363893193455880025]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 185825674244197562991939767988017108291227572248603957165920198900856248993890
discriminant (Δ): 4941992565371643620810087164829754113722511270106186147330892577260865813848839664661473374681600000000
Faltings height: 18.4180
naive height: 244.8592
primes of bad reduction: 2, 3, 5, 7, 13, 31, 41, 109, 557656420008563, 2792657071781563542343, 45070114994698512690168645973549
regulator: 168096208662763960510.892493175875190472772459429509476031134792547891704698269
submitted by: Bhavik Mehta
submitted at: 2026-08-30 20:00:34 UTC
last updated: 2026-08-30 20:00:34 UTC

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

(81994795736121090, 14389357183593365506103205)
(19638448141141470, 5194338555763404445088325)
(25407788636088590, 1472000573416426419118205)
(23922877357374090, 2848634662225121255693205)
(2082271354351501590, 3002734786820423758458380205)
(594748118308858614/25, 367912189195286588035045977/125)
(296103931252848735/4, -83103738215046850691336235/8)
(-81275952408710874, -8424675073539321827453043)
(76687090186005030, -11735889219566931404833395)
(64654840840903590, 5213234711425779425048205)
(13089879709325670, -7735410088313394039061875)
(-33053018863096410, 17045669641615903274648205)
(60197740356770150, -1018681973554748129450995)
(-85889992228344690, -1074098136462964401309315)
(-85682174929833660, -2128377329845640170635795)
(60860515824480870, 2111337181981886085512205)
(-81152299857542910, -8526478603122817094960295)
(20668073488375590, 4715078211635330525648205)
(-6044247061084410, -12998042655830195586051795)
(-44030040743680260, -17473314403055424696393795)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -34. 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 20:00:34 UTC · history