curve #3085

y² = x³ + x² − 6692387808676988977764474943486x − 3498869682258563508267771748897941752628277240

a-invariants: [0, 1, 0, -6692387808676988977764474943486, -3498869682258563508267771748897941752628277240]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 734580525126326665789875840301734560160
discriminant (Δ): 13894715478174642653334291858041720100534300308598766793298204538999900808579987262355881960000
Faltings height: 16.7613
naive height: 224.5492
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 163, 199, 229, 313, 1531, 18541, 53173
regulator: 3119618200.250326330542250799162208189448449411595457314255
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:33 UTC
last updated: 2026-09-26 20:36:33 UTC

Witness: 10 independent points log in to add more points →

(-568634665484218, 11081002979887647935700)
(-15825575908682863/16, -2969565728584437386150175/64)
(-166008412224986299/196, 108443708345701496526295041/2744)
(-91060078215418579/100, -42897078946186349308521669/1000)
(-19554463788667306/25, 4432185394671500279504028/125)
(1705570539314576966, -2227429816741713780114063444)
(-4175562463625947/4, 387790920387792066063375/8)
(947318146534941175478512017495677/154311956729267236, -26197572499128993356363772657875602671668921016675/60617687915220179917645016)
(69937837990825890665/21904, -284170090010106145902351934275/3241792)
(11995192851768521, -1281463535056311742707234)

Commentary

2-isogenous image of D-z4old9101112-031, rational kernel x=4539034041928485. Source: https://dujella.github.io/tors/z4old9101112.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 20:36:33 UTC · history