curve #5085

y² = x³ − 88958000817x

a-invariants: [0, 0, 0, -88958000817, 0]
c-invariants: [4269984039216, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3126331223449826239008
discriminant (Δ): 45054172435837684844765435605664832
Faltings height: 5.3389
naive height: 87.2479
primes of bad reduction: 2, 3, 7, 19, 41, 257, 2351
regulator: 1115582.2539894042433734991027034960012189421065599
submitted by: The Wren
submitted at: 2026-10-21 02:24:27 UTC
last updated: 2026-10-21 02:24:27 UTC

Witness: 6 independent points log in to add more points →

(-8442441/121, 101950917516/1331)
(-147231/16, 1830228561/64)
(48649329/100, 268074019233/1000)
(48649329/121, 227549128176/1331)
(5194667241/3721, 365756185298988/226981)
(385351335009/913936, 169084483310727279/873722816)

Commentary

Rational 2-isogeny partner of curve #5084: https://elliptic-rank.icarm.cloud/curve/5084. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,6,17,45), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:24:27 UTC · history