curve #3188

y² + xy = x³ − 393744130919862680997061205x + 4952716613400599790174081417550462849761

a-invariants: [1, 0, 0, -393744130919862680997061205, 4952716613400599790174081417550462849761]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 300953424448851386334317567807295829194402996102
discriminant (Δ): -6689891930585676000307597032645591964221040103265199283004176232615267149826490368
Faltings height: 14.3967
naive height: 196.3247
primes of bad reduction: 2, 3, 23, 29, 71, 73, 101, 257, 307, 353, 449, 1181, 3511, 4217, 16561, 204751
regulator: 358410078857257.841950288637918180941690802449631611601764979796683
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:46 UTC
last updated: 2026-09-26 20:41:46 UTC

Witness: 14 independent points log in to add more points →

(-6372209542377614/361, -548852388221684244189251/6859)
(107675046974519266/15625, 98948474849875970557117811/1953125)
(-1627690600873586366/85849, 1882735939282522808454741947/25153757)
(-639563558500833998/117649, -3359916083270508286819461887/40353607)
(7857406760540602882/11449, 22016109232939843856176493717/1225043)
(-8022914243186, -87150898802595089729)
(1602997543139990818/72361, 1640340770295025500677903747/19465109)
(872227104966370/169, -121493063200393358730173/2197)
(8847379658179771410340/576816289, -695696047182503994718169165634793/13853396812913)
(1540298477232754/49, -52734276279490976627087/343)
(3487630933315490662/1338649, -97274644568460341566690379281/1548816893)
(2297186214097913026/151321, -2927794232721542010368260453/58863869)
(8400705059447479090/22201, -24316191262061460024867995749/3307949)
(1039385118855579298/100489, -1419844613583190835327695693/31855013)

Commentary

2-isogenous image of D-z2old141516171819-054, rational kernel x=195391032383343/4. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:46 UTC · history