curve #8

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y² + xy + y = x³ − 940299517776391362903023121165864x + 10707363070719743033425295515449274534651125011362

a-invariants: [1, 0, 1, -940299517776391362903023121165864, 10707363070719743033425295515449274534651125011362]
rank (lower bound): ≥ 22
conductor (N): 22720638514787473197194583889675055980109503436060704437972911338086049759883790 ★ record for rank ≥ 22
naive height: 239.3849 ★ record for rank ≥ 22
Faltings height: 17.8713 ★ record for rank ≥ 22
discriminant (Δ): 3680432205940761304587473756368167508478483727616134375685404923405185646725374901862332174937532900 ★ record for rank ≥ 22
primes of bad reduction: 2, 3, 5, 7, 13, 17, 37, 47293, 270704849145149791, 60794657878864337775664712674231370427122734380997
regulator: 12992022722213680254541.35531967258530327885791487307445312157323896249403008828342
submitted by: David Renshaw
last updated: 2026-07-01 22:29:41

Witness: 22 independent points

(32741153161482344264/3025, -223089674587110979578532169697/166375)
(215521674613198983365/24649, -6872949155061353554235704378947/3869893)
(637312541911044643/81, -1420356190129296832193564087/729)
(-11906250919327880080/361, -16580788535875788634285886853/6859)
(-136152345735493381/4, -14482270545045735913281693/8)
(-27830298157016213012252/7134241, 72099692861364392796183359497454267/19055557711)
(4127671322151440, 2626107692045613116291646)
(6175679781777296, 2266254335997033124678449)
(12047255022287093, 1061993236525943920980477)
(416685837455186583191/32761, 5321268222786709669160311587369/5929741)
(149915813139075767108024/10220809, 8704326838108646949177663157917117/32675926373)
(58759417448623559/4, 2030968553150713398654657/8)
(237195157887349854919517/16024009, -11477798111611307979707215505421441/64144108027)
(9568474434078537574436/687241, 319520556343135681977874272805086/569722789)
(1725892668710258675291/177241, 117378050663464845770966453025039/74618461)
(-35277008506980340471/1024, 48766027143946934186731674507/32768)
(-2752742763529705669/121, 6000532252185982381233585699/1331)
(-18552633109178014, -4665466215824339436717966)
(-113251707338691187737649969/3304065361, 310152527894831470820009872373229341739/189920981015641)
(-7572001778163591251/729, -86590661426506799357663502953/19683)
(-380526048554032285152211/11242609, 73081235744931307684790623068490233/37696467977)
(-1503889497722021588110681/42784681, -160705885170116750151534640924719585/279854598421)

Commentary

Found by Fermigier (1997). A historical rank ≥ 22 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:40 · history