curve #398

y² + xy = x³ − 12892599774455576272301592959047823530919513428112484011550x + 560755046348395412977088824999503890617558856687636981223935662848386484980312656296132

a-invariants: [1, 0, 0, -12892599774455576272301592959047823530919513428112484011550, 560755046348395412977088824999503890617558856687636981223935662848386484980312656296132]
rank (lower bound): ≥ 30
torsion subgroup: trivial
conductor (N): 2835647668537242470520670933169702570972603599817629325681756626410520535791722336193301955569324372206950385073317702302276416849610 ★ record for rank ≥ 30
naive height: 413.0256 ★ record for rank ≥ 30
Faltings height: 32.2529 ★ record for rank ≥ 30
discriminant (Δ): 1311021171349839305076869751181166178696551783990116503463598327754948038424504439658681699321482006153838737418363257993512584919966486940354703657840019569754443427840000000 ★ record for rank ≥ 30
primes of bad reduction: 2, 3, 5, 7, 19, 29, 31, 37, 41, 59, 73, 79, 101, 151, 197, 20084070565614383, 277131689105980733414153, 91586120381369539248864736998169886452607476058975046580346704317125237
regulator: 29644133919811878908161048367470898.683586921990156088039577443431947190892218978646226017207321353
submitted by: NDElkies
submitted at: 2026-08-28 16:21:59 UTC
last updated: 2026-08-28 16:21:59 UTC

Witness: 30 independent points

(255988718874926192236548018804, 118471064338162161016351388941923332809406898)
(1066205451814910034394295004, 23388247010770888011772107443289505784724498)
(113917476844419910281731337891/4, 117760659397165502491412133574198971289541859/8)
(522896578822997794950102133404, 369851424817590721242251565943695349358330898)
(752250152281037640723473915004, 645403618239626742448521759727984493932740498)
(61804854104619698852310920004, 120214803870231627655588231658347056480498)
(20899766145823995773278515804, 17332967103311289782942907345374643157361298)
(-93254972898947099650813769268, 30855520614764535640323270586626377097290850)
(69713966164712304592492810404, 879341993008471697868322219521408586366698)
(60167736519416976225848653404, 1689119177236660096430130798955079194250898)
(-14046491075437784120213373996, 27186014904486995598713981999311252231560498)
(-130496575431865945133637499386, 4574111880086616610004422937977208277497508)
(42793421047455475252311947484, 9348967782365268719193346216572636318501138)
(307730699417145077975860872144, 160420835618426919369879316139943794781559218)
(41901340123969117728348005179516/729, 61004412435011873236122293828203075348860441334/19683)
(58130399340913205751269159004, 2781011322605723291068350540205198693092498)
(70345925383762050501865122204, 1386888123108366471979074557284674228142098)
(47475994387288306853069249004, 7461603107396365118336363485188332292162498)
(49620349750437331732008503004, 6572208525446476744123366284203963791284498)
(61805127635300558534355248604, 118573275438486609948049447535048685070098)
(59748726488260912696099828854, 1932931405414328200984836878686914954536598)
(59841577032822742951082534364, 1880183337056779424275360836669144786159378)
(-118760392546072421990801612496, 20417762309697775621233399179218700962920498)
(81538823240248716448164032604, 7185053709895791307497392456302587466433298)
(69804093051086399717838542304, 962825039084755225856791363805867260214898)
(60968339684582708695209678044, 1158621242686468864160533635604978093990418)
(84118386438376926037642559004, 8453631999329921882625980842970149886292498)
(-100777109571234949083590688996, 28922984981811818231788164913656183379020498)
(52934781599204816902071863004, 5159086508060990188972690790046932481972498)
(52008008801558252504808621504, 5559638586120196928159633060055634855729998)

Commentary

Noam D. Elkies and Zev Klagsbrun, September 2025

last edited by NDElkies at 2026-08-28 20:43:18 UTC · history