curve #5238

y² = x³ + 8689468788x

a-invariants: [0, 0, 0, 8689468788, 0]
c-invariants: [-417094501824, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 16779303959472486432 ☆ record for quartic twists, rank ≥ 7
discriminant (Δ): -41991332555577884485137835511808
Faltings height: 4.7574
naive height: 80.2697
primes of bad reduction: 2, 3, 7, 11, 13, 59, 61, 67
regulator: 53156.36711646379883269099189319490139822263749481679
submitted by: The Wren
submitted at: 2026-10-21 10:51:42 UTC
last updated: 2026-10-21 10:51:42 UTC

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

(7198, 7932196)
(8437, 8597303)
(17446, 12526228)
(28548, 16472196)
(39589, 20150801)
(64782, 28892772)
(356301, 219837717)

Commentary

Rational 2-isogeny partner of curve #5237: https://elliptic-rank.icarm.cloud/curve/5237. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (10,13,18,19), reduced by rational fourth-power scaling. Reference: “High rank elliptic curves with torsion group Z/(2Z),” Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Seven rational points checked exactly; PARI/GP returned rank bounds [7,7]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:51:42 UTC · history