curve #4758

y² = x³ − 1577047042x

a-invariants: [0, 0, 0, -1577047042, 0]
c-invariants: [75698258016, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 159172951851580784896
discriminant (Δ): 251023232883943899844275077632 ☆ record for quartic twists, rank ≥ 7
Faltings height: 4.3307 ☆ record for quartic twists, rank ≥ 7
naive height: 75.1501 ☆ record for quartic twists, rank ≥ 7
primes of bad reduction: 2, 41, 2593, 7417
regulator: 1001049.564428045052089109962758498520935217667868105
submitted by: David Renshaw
submitted at: 2026-10-20 21:28:45 UTC
last updated: 2026-10-20 21:28:45 UTC

Witness: 7 independent points log in to add more points →

(-7417, 3359901)
(-23337, 4908549)
(-304097/81, 1765891279/729)
(1498754/25, 1374533742/125)
(-289, 675087)
(-21689/81, 473715107/729)
(1201554/25, 741892842/125)

Commentary

Quartic twist y^2 = x^3 + a x with a = -1577047042. Source: Aguirre, Lozano-Robledo, Peral, 'Elliptic curves of maximal rank', Table 2 (Rev. Mat. Iberoamericana proc. 2008; preprint https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf).

last edited by David Renshaw at 2026-10-20 21:28:45 UTC · history