curve #4986

y² = x³ + 604907060241651475x

a-invariants: [0, 0, 0, 604907060241651475, 0]
c-invariants: [-29035538891599270800, 0]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 468368065958652117289244717273184800 ☆ record for quartic twists, rank ≥ 11
discriminant (Δ): -14165957494505808390700888235053621088485554243379000000 ☆ record for quartic twists, rank ≥ 11
Faltings height: 9.2720 ☆ record for quartic twists, rank ≥ 11
naive height: 134.4452 ☆ record for quartic twists, rank ≥ 11
primes of bad reduction: 2, 5, 7, 11, 19, 23, 47, 73, 229, 613, 1493
regulator: 234917979030.095295349434557145588566038938178883021915075575
submitted by: David Renshaw
submitted at: 2026-10-21 01:12:08 UTC
last updated: 2026-10-21 01:12:08 UTC

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

(2707655531, 146590523489746)
(5584119209, 421312284729902)
(14728762475, 1790004158911150)
(64057253771, 16213787730992894)
(6483340556/169, 10596394496418346/2197)
(8685878355/361, 26179787467599150/6859)
(9431228745/4, 964453170178275/8)
(114579590755/9, 38857049242483850/27)
(1860950287449/38416, 40838013762168182307/7529536)
(2315212172009/71824, 85071089078804015627/19248832)
(330753855368475/4133089, 58770365094917229565350/8402569937)

Commentary

Quartic twist y^2 = x^3 + 604907060241651475 x: the curve y^2 = x^3 + 4Bx (2-isogenous to y^2 = x^3 - Bx), reduced mod 4th powers, where B = B(89,102,168,214) is the parametrization of Aguirre, Castaneda and Peral (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/). Found by a Mestre-Nagao sieve over all quadruples a1<a2<a3<a4 with a1^2+a2^2+a3^2+a4^2 < 3*10^5.

last edited by David Renshaw at 2026-10-21 01:12:08 UTC · history