curve #4770

y² = x³ + 14705318209188280854082275x

a-invariants: [0, 0, 0, 14705318209188280854082275, 0]
c-invariants: [-705855274041037480995949200, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 30755041227873340789321920108558919243453506727200
discriminant (Δ): -203518200506661963938068936541957059295633128584811935369651194232651171000000
Faltings height: 13.5236
naive height: 185.4644
primes of bad reduction: 2, 3, 5, 17, 23, 41, 47, 149, 157, 433, 967, 1033, 8573
regulator: 6223581569868.4046405347561169735024187140823387390060623396253
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:30 UTC
last updated: 2026-10-20 21:29:30 UTC

Witness: 12 independent points log in to add more points →

(70283712595185/16, -782160378242464718775/64)
(1512279300385/4, 18954559846587001175/8)
(2019469851315, 6158966376302652150)
(3712356096535585/6241, 1475635708760395281769450/493039)
(157351802491843065/90601, 151293680993523885994398150/27270901)
(2335396509025, 6861499898380881850)
(1896413333936326960/77841, -2643710532687591534844576300/21717639)
(1021486650468435, 32647663331556118244550)
(6567622476950065/6724, -2156346771080285891554975/551368)
(709340625665945100/208849, 901053046039007942274748950/95443993)
(49086645564000960/9409, -13497322809659653028071800/912673)
(4120354715760241/576, -300092110267293249137911/13824)

Commentary

Quartic twist y^2 = x^3 + a x with a = 14705318209188280854082275 (2-isogenous model of the curve listed there). Source: Aguirre, Castaneda, Peral, 'High rank elliptic curves with torsion group Z/(2Z)', Math. Comp. 73 (2004) 323-331, doi:10.1090/S0025-5718-03-01547-3, https://www.ams.org/mcom/2004-73-245/S0025-5718-03-01547-3/.

last edited by David Renshaw at 2026-10-20 21:29:30 UTC · history