curve #339

y² + xy = x³ + x² − 491829755436006895005968707827x + 132444599116949764711601499902718391049688249

a-invariants: [1, 1, 0, -491829755436006895005968707827, 132444599116949764711601499902718391049688249]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 1238318316451373978295596809992671091823834684624403142859416848506306975530
naive height: 216.7174
Faltings height: 15.8659
discriminant (Δ): 36242601339908124690239703627094837632163784603962972713449076762616483455697692209362500
primes of bad reduction: 2, 3, 5, 7, 11, 23, 43, 847414652347307783, 48633691997858641753, 13151970204492846993007887537533
regulator: 46569763744352744409.61359720185383356072702033688728301781298292503329077965
submitted by: 7fff-zip
submitted at: 2026-08-24 19:18:56 UTC
last updated: 2026-08-24 19:18:56 UTC

Witness: 19 independent points

(1325755309181584563, -1526493649607651770438588644)
(14811900568473668, -1800684850039588782758819)
(3479117997450980808/961, -6375690841885100515680494979/29791)
(126763995381026907563/63001, -1350053075773475369221449805644/15813251)
(372654799349919293/961, 5549544721805604674024221/29791)
(182368752388882452/529, -23909733342456329501142423/12167)
(22191720064133473282/71289, -58688533311249753481988450747/19034163)
(38735597402733, -10651356759778027021344)
(32848283322701439923957/21003889, 5435396622374262026009925467739197/96260823287)
(1607839740652035, 59145293836504729834497)
(2787903644014973, 142933520469638124141696)
(3395145223101368, 193903293218894149902451)
(46678484992606296753/10816, 314942374992992739399337786779/1124864)
(7796853309458321021670622877/616771481104, 687422877097012406755920748397786591849627/484380249142064192)
(98137870007750233836620148/10730680921, -969420302142226156410687178371464147661/1111580505925469)
(44708850599265413, 9452292275726751604748331)
(9301924261107281490192/24275329, -59344901259662282399619011214357/119604545983)
(356596060653093, -1550828506955649990069)
(27192694498319271858751683/10532711641, -137034638259707168699920347357655106916/1080961663004189)

Commentary

New in this search: Mestre quartic family with roots [0,113,550,753,868,1058], T=6141/11. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by 7fff-zip at 2026-08-24 19:18:56 UTC · history