curve #5764

y² = x³ − 4076999032604998370002809075

a-invariants: [0, 0, 0, 0, -4076999032604998370002809075]
c-invariants: [0, 3522527164170718591682427040800]
rank (lower bound): ≥ 8
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 16307996130419993480011236300
discriminant (Δ): -7180669920324423986823780244721240642127926904869817630000
Faltings height: 9.7804
naive height: 140.6735
primes of bad reduction: 2, 3, 5, 7, 19, 31, 163, 733, 1663
regulator: 16699100359277.6310266998760248657077453859458918753010
submitted by: Maksym Voznyy
submitted at: 2026-10-24 20:13:21 UTC
last updated: 2026-10-24 20:13:21 UTC

(109042271981300576431438805301159395149711/45527731474991951425028045767129, 30195945621686953024679036190989575802332922927167269514539766/307194910851189057907896487891843457412588895133)
(15953778582451670641/124367104, 63722810344176620330453262089/1386941943808)
(599445710401390819/31326409, -463979078262532929776685328/175333911173)
(1795288996223541370037892420820681/321903222906614015716, 76067922173597209399755359298358397530920697492571/5775478710722611108750710554264)
(60122809937954748285572857891/15108067070370825849, 14257266453036197932083233880811780174474036/58723692549831415082084616243)
(835153589800032274096882675801/521328121515433690084, 69583335282984278428423026549836674353234301/11903282161259034463778701497448)
(773579636129789244620463211/13655919539517025, 21515560926514681727598730303489346225584/1595810888025029544228625)
(792812690699527530955283703640557256519/69044056641871205321908128769, -22293094775087844465306386081470582281720591458739553939322/18142179110730118397548456912881139430895103)

Voznyy (2023), A = -24576408539130, [r, s, t] = [7, 43, 413] for S3 in https://arxiv.org/pdf/math/0403116

last edited by Maksym Voznyy at 2026-10-24 20:13:21 UTC · history