curve #458

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y² + xy = x³ − 24459647223008819321132521677571770x + 1475866115712309323058485904595825308688025746908900

a-invariants

[1, 0, 0, -24459647223008819321132521677571770, 1475866115712309323058485904595825308688025746908900]

c-invariants

[1174063066704423327414361040523444961, -1275148323975435256883626421627428057827995806114457041]

rank (lower bound)

≥ 21

torsion subgroup

trivial

conductor (N)

127539526964771188715253520186359953250470657797397956136339909486451272206611990

discriminant (Δ)

-4425023236897122347983043266683538093006074579543760499341768527284100218962279072060350429538560000000

Faltings height

18.5694

naive height

249.1653

primes of bad reduction

2, 3, 5, 7, 11, 13, 83, 97, 5541979, 13598052927397543826778726821571445157012029493876346741264068411

regulator

2255884186515467036914.3517382021156105599407290746853899506328130897926054055134

submitted by

Bhavik Mehta

submitted at

2026-08-30 15:56:56 UTC

later contributions

rank ≥ 20 → ≥ 21 · Jason Ma · 2026-09-03 16:33:17 UTC

Witness: 21 independent points log in to add more points →

(135790045638840660, -25657621374695949238556490)
(23799517271612940, -30120072569537908153436970)
(279755952459059052, -128560430458044660320249538)
(69815632498409172, -10416203940294171765134538)
(-543810692566283540/9, -1411559441052274594044751190/27)
(29679615187438140, -27857806188535086843001170)
(65083491858257940, -12634535402978304980600970)
(108696330510912120, 10071129561251200638488850)
(21233015442698089020, -97837594353090533745023348970)
(3642702655023805440, 6946108816893288784373695530)
(23361806407705140, 30285224751988812341791830)
(61946169258468900, 14085184793774728314043830)
(-174803064575155950, 20253036826296762282908580)
(296778477490662435/4, -66703654440583699926003885/8)
(97900766121927990, 4425345266394687227887530)
(90364581538404240, -1864295619901145456962470)
(196795348131855540, -65451448257662630973846570)
(359795922537300, -38302293416520773909957130)
(-1033073548019628, 38744465769370427027570166)
(-1437242866184520, 38871809841884083084297590)
(-1463681124897708, 38880124793983528544083446)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -40/9. 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 15:56:56 UTC · history