curve #403

y² + xy = x³ − 4789549597347342217921487845151485156552215088671735288x + 4038895070100529332977411476954657751699524878306148134823129956225541669799702592

a-invariants: [1, 0, 0, -4789549597347342217921487845151485156552215088671735288, 4038895070100529332977411476954657751699524878306148134823129956225541669799702592]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 23041381240412411159163335278781148219756014215360760109370073705697070765442268382401335882318018176926510969767537593177464755040274350
naive height: 389.3339
Faltings height: 30.2210
discriminant (Δ): -15315564775743859773341691289544645120730310762127768915894882508509352445128944124435441714677299462641137372755597315156825505771232278317103481761930549022720000
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 47, 5076671, 16143317537697055815553, 14500332007970721183436484998398890113459, 506635894750112623868634415118765751634240595748702020977
regulator: 20740304673472845962849215040060774.20396559799586898592951377474738348070256530816221888161347
submitted by: NDElkies
submitted at: 2026-08-28 17:01:42 UTC
last updated: 2026-08-28 20:23:57 UTC

Witness: 28 independent points

(1419451495537511168684583312, 10016636136282285835078838529466974857064)
(1292140875431703451802599632, 2741696599900168568705766212407537728264)
(-1554127539571468469351952528, 87913380492337621277952909907301690306184)
(1397958835686337759990146672, 8678556074583426665448039541184765320584)
(1890138070783235545861300392, 41698126460034390763348625554299939764424)
(1276927435871250956150229552, 2252496318802875130984125648714648599304)
(1217640884954401260543942432, 3504011368921423189906245824679832351704)
(1202874839383604410442991252, 4256224887588367893909620787164335591704)
(1724087135177671081546340292, 30101538707170661239481514422736376344984)
(1252832940868607213371207632, 2196381457007702314449378309397226964264)
(1342918522827768584938031472, 5364696161567427886286699934055105394184)
(-1968110703356380069775016048, 76432069985520801134007772286352685216104)
(1186623113932211878168758192, 5134048983724854708597634990994257566024)
(1246645161819603976729700172, 2338290308192325124454271360011026845084)
(409944476705728968197334912, 46307004547414160530881672265476470592264)
(1319128590833893794585362412, 4034753450891939075667566276312453322264)
(1273264904321123077970197272, 2179724916324025019958356503780580579784)
(1186696906354756081579279512, 5129986505942008589202952547315553538464)
(1242947307908468956781739432, 2447237983284829504997402614270782654864)
(1278669687560602602599205552, 2294171880161698309278818529944665923144)
(1094909451213329591541975408, 10362366528276628104240321013380668257800)
(1193773363028132974541505732, 4743269166135887844505620610174208876964)
(1326288578262716370649729062, 4423420861494276123455035681945813451124)
(1236406574312796280548185712, 2675947459623539614255135459247126649864)
(-107183062476078148811179248, 67461265348457412006093827698026366113544)
(1456689479851038650097837252, 12370222742162239949637704552167747191204)
(1012511438842683902742601662, 15080689613131735052862433148397828962574)
(-1606532526745840653615624288, 87103888361989004911611532588079598044424)

Commentary

Noam D. Elkies and Zev Klagsbrun, January 2026. The cubic field generated by the x-coordinate of a 2-torsion point has a class group of 2-rank at least 26, the highest known for a cubic number field.

last edited by NDElkies at 2026-08-28 20:42:05 UTC · history