curve #5582

y² = x³ + x² − 1813949458099361403555204x + 936688741703516610350327598008223456

a-invariants: [0, 1, 0, -1813949458099361403555204, 936688741703516610350327598008223456]
c-invariants: [87069573988769347370649808, -809299072832360768786615660763328964800]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 8224704040914907186872
discriminant (Δ): 2962436917453602713282171144417593825485680559381630768222592295957915904
Faltings height: 12.7577
naive height: 179.1862
primes of bad reduction: 2, 3, 7, 11, 17, 23, 29, 41, 61, 71, 73, 101, 409, 733
regulator: 9793538.888146978564732718527697132134261874164444370
submitted by: Daniel Błażewicz
submitted at: 2026-10-23 19:34:59 UTC
last updated: 2026-10-23 19:34:59 UTC

(1594526884956, 1448588160684053928)
(41802139063116, 270131365570434670512)
(-92004433010639046/143641, 73761077402996214674222010/54439939)
(13709797205625/16, 6864216365024910549/64)
(99532310090184/25, 942246766872206378352/125)
(138966918817340/169, 72747794041461774312/2197)
(217291533444581832/5331481, 11434930952640744257936987760/12310389629)

(u,v) = (281358,1539919) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 7.

last edited by Daniel Błażewicz at 2026-10-23 19:35:00 UTC · history