curve #435

y² + xy = x³ − 491207823564768633438000932167842220x + 134132256298970127093936962644107344568006870419584400

a-invariants: [1, 0, 0, -491207823564768633438000932167842220, 134132256298970127093936962644107344568006870419584400]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 123075360516683336654553303844383299197909575873281085669191642645264530494551661964770
discriminant (Δ): -186958601393353252327110498841398254188347817102394513204661439461977075991984262361321535370901510400000000
Faltings height: 19.3887
naive height: 258.1845
primes of bad reduction: 2, 3, 5, 13, 37, 79, 4007, 17669, 8512039, 1766124707838611, 101435835322593306728071336404869210405812124818463
regulator: 3705880620238679320250.29168292770768152586822708798026324741640090924409721055
submitted by: Bhavik Mehta
submitted at: 2026-08-30 04:29:51 UTC
last updated: 2026-08-30 04:29:51 UTC

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

(40838553511767320, -337846339852693382537605420)
(-809018868370014760, 44905200779902088460562580)
(380124162765758840, 48353888062591115809534580)
(695649576180394040, 359260742931272127389907380)
(366425412797439590, -57795167443891501590647920)
(378712735020910976, 49212210585677384124657812)
(418738359956518040, -43209276172932262877821420)
(363793068240626840, -59838055638053255769205420)
(-35355377301999160, -389172069788933719229005420)
(1103477090798054840, 967345571278400532217090580)
(1853400678503207090, -2364388512209569933886249170)
(163561244489712604370/289, -950248241338883047220401741280/4913)
(785031764540957840, -481989483160081856190685420)
(325041624449940872, -93864710367607361756902684)
(37873571236452314390/121, 139897997056148676557614114480/1331)
(4933239588345764765/16, -7008159686299436628002309005/64)
(1708850342053651640, -2069991361572707163592189420)
(112051284482741432, -283722835834304811640221388)
(-786318334054936660, 184932642990043922970432080)
(42714225031433289140/121, 91459006904937138737703223480/1331)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 40/3. 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 04:29:51 UTC · history