curve #328

y² + xy + y = x³ + x² − 173863764473375398357582442830175x + 883145447815390783031613032285356595588099614085

a-invariants: [1, 1, 1, -173863764473375398357582442830175, 883145447815390783031613032285356595588099614085]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 43849915595748976027625327568140150988792878346340410934428630268647228784770
naive height: 234.3228
Faltings height: 17.2945
discriminant (Δ): -574401154490032448091180309990921523509969094823336020555771703114693561881267196456934604800000
primes of bad reduction: 2, 3, 5, 11, 17, 23, 61, 73, 2191393, 4871525053, 3391795766524988728488283, 2107708103066596881109715329
regulator: 215082836065331596042.663713546393845550767010246472599703982535459336787726353
submitted by: 7fff-zip
submitted at: 2026-08-24 19:17:24 UTC
last updated: 2026-08-24 19:17:24 UTC

Witness: 20 independent points

(31941998887847240931/25, -180517827928710470478646631686/125)
(206258866031062743, -93487086643226687096754002)
(771254121484677847/9, -669738464562928802228822734/27)
(-5933220306606892893/1849, 94342751202953563221961862086/79507)
(-1600978306397760897/2401, 117577254716661291274191781642/117649)
(464778094526026998139/65025, 1240061708875631545263726393502/16581375)
(313311597703542413061/42025, 312436274991260495582176776376/8615125)
(3093411392862482313196323/394459321, 347429895526098239700399623257167418/7834356574381)
(8185285099881883, 91793246312288872654658)
(15096858470837763, 1303514990142403131708178)
(10797960552934920996148563/687435961, 25665349393729466350838506786680856022/18023883461459)
(541041732849712921874763/74839801, 41227696227699087914184048091151678/647439118451)
(570676175961381813, 430992475393457730886552378)
(11424309699197275467/529, 32648714273828998906473039086/12167)
(723652538888165559201641715123/86483322728881, 95814719273851855249610555828360250364453018/804263853865733631721)
(13540405013375301957/2809, 59001940533819441962550391016/148877)
(34878168740500992225802147/986148409, 193335354761655581913272612311002621766/30968018487827)
(2567487415508159083/81, 3803342018541561798404623442/729)
(8409094814293764983571/1155625, 71315186350211886547018194600406/1242296875)
(3684580453317449370054662715/757288069729, 257261432393715538613612283629233370467870/659009495903779567)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=17595/22. 20 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by 7fff-zip at 2026-08-24 19:17:24 UTC · history