curve #3121

y² + xy = x³ − 1894034708782161397859898445391439660840x − 24965016703598553227577391083076701696774653385041938046400

a-invariants: [1, 0, 0, -1894034708782161397859898445391439660840, -24965016703598553227577391083076701696774653385041938046400]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 31000471439531693917957022221199714058221545530
discriminant (Δ): 165609415571190820129341365232069571270124372402764583095561961310796627580712177957014655281457550022615611022828160000
Faltings height: 21.5895
naive height: 282.9322
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 67, 71, 73, 103, 127, 293, 1091, 682327, 102838432973
regulator: 1689537981991.91455143431322266739176514950284828901413625782
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:16 UTC
last updated: 2026-09-26 20:38:16 UTC

Witness: 11 independent points log in to add more points →

(270520363505805001260, -4388585534829895271467133883540)
(216829221525603796410, -3123869320474969790700763756620)
(49955574794449754640, -71305663538328353947751925000)
(1022983896738424713360, -32689234897974033976782513476040)
(5922447628031699030160, -455764076888700953850257694834120)
(47695513110379260585420/529, 8917679408948952181437551485480920/12167)
(69788135118789775560, -427491328683066413936041976040)
(1044022590661252763618352867390/2796188641, -1059226367827156612544574501353960335960279710/147859659147439)
(3138816300197070173811648/6889, -5534784702807214459827258354138681336/571787)
(2200922517635020411958922467578560/10790916212209, -100720506600816079858020777783316362601522074032280/35447652584044591177)
(193800494264237537790820/9, 85316186113891562684060478449757020/27)

Commentary

2-isogenous image of D-z4old9101112-004, rational kernel x=17040500771125811280. 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 20:38:16 UTC · history