curve #4766

y² = x³ + 178992718735846748616963x

a-invariants: [0, 0, 0, 178992718735846748616963, 0]
c-invariants: [-8591650499320643933614224, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 113914287503822021953053632003640671122813665312
discriminant (Δ): -367016904416987949210085997608735791527509255137467429227321669821974208
Faltings height: 12.4214
naive height: 172.2385
primes of bad reduction: 2, 3, 19, 43, 47, 67, 109, 127, 137, 233, 593, 29501
regulator: 1547116916961.9386157068588757311731399190223531028116272994743
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:13 UTC
last updated: 2026-10-20 21:29:13 UTC

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

(1815169595587, -2511094727517482278)
(3108488718201/25, -19436677436460539574/125)
(23105147088064, 111080007661807727576)
(1355857371084, 1653852449601640314)
(1976996525091, -2842708052312298198)
(52048932822297/121, 526693873308984736182/1331)
(250967178729/16, 3393473035525650549/64)
(127512964313347, 1439905210956249140422)
(3811061429424, -7485634881031236156)
(101099605527603, 1016548243412394901554)
(237683398557369/196, 3880966074347686800069/2744)
(845515222401/16, -6272780001797684673/64)

Commentary

Quartic twist y^2 = x^3 + a x with a = 178992718735846748616963 (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 21:29:13 UTC · history