curve #4645

y² + xy = x³ − 421998609688798141378661266913244944813586572790972472x + 105133199416296112543076668482970837293052409636073661102540207419654670363630656

a-invariants: [1, 0, 0, -421998609688798141378661266913244944813586572790972472, 105133199416296112543076668482970837293052409636073661102540207419654670363630656]
c-invariants: [20255933265062310786175740811835757351052155493966678657, -90835084295679841237218241599670703318790748104831254410348375237159868435126904769]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/3ℤ
conductor (N): 2066118591514886972068227354856277642466926416765374242203646246578
discriminant (Δ): 34753618831154587683475807314832683743831573461204385226998114851574443691424060983243005393065318230806135597106228113105370583096363685752938310654259619692544
Faltings height: 29.6597
naive height: 382.0441
primes of bad reduction: 2, 7, 11, 17, 19, 23, 29, 59, 61, 79, 101, 103, 109, 131, 167, 293, 397, 2777, 8963, 22039, 30223, 37783, 97453, 113539, 576391
regulator: 7928098966897021.7328744385089189444439876394908978471246568321090129
submitted by: wgxli
submitted at: 2026-10-20 01:08:10 UTC
last updated: 2026-10-20 01:08:10 UTC

Witness: 15 independent points log in to add more points →

(356343248488963996817346160, 74105837486973349787674899791447806664)
(393337461706575748426283282, 17047298404861392355967753865722399460)
(393352761077877796080546416, 30588730548740733767502094334027430600)
(352053498545987401950598160, 448563894961298117409492842382940862664)
(354497790513135295319149744, 291368000806436319496368157586526918344)
(355553899935888251350589536, 196381286263365703730360622272001941720)
(356130952392846157784329520, 119398926383371972853824496319768974344)
(393568217281674489427587424, 100391866055138572108043910648619966424)
(394078199009744146651980982, 179322219641743968634426996128341233460)
(401151829801907680239776766, 634158381720603690305492744852020787700)
(402262493181987326376684644, 686394582438115881169323519374306572612)
(406785098975032665480946416, 884792724825890274905662949846679222984)
(243918530718453398530682330, 4088050436119478802644627509025957361346)
(320617115404726784246408000, 1670678352820427100133420610563356340216)
(445353265742356230110015920, 2350722962453393371319406248024344782536)

Commentary

specialization at T = -5652311/1171368 of 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.

last edited by wgxli at 2026-10-20 01:08:10 UTC · history