curve #1750

y² + xy = x³ − 1144386x + 471132612

a-invariants: [1, 0, 0, -1144386, 471132612]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/7ℤ
conductor (N): 6378
discriminant (Δ): -10662541623558144
Faltings height: 2.1330
naive height: 53.4649
primes of bad reduction: 2, 3, 1063
regulator: 2.883891156379270148166181353646264737736
submitted by: hdeping
submitted at: 2026-09-25 10:47:49 UTC
last updated: 2026-09-25 10:47:49 UTC

Witness: 1 independent point log in to add more points →

(-774, 30276)

Commentary

Specialization t=-8/1 of the universal X_1(7) Tate normal form family y^2+(-t^2+t+1)xy+(-t^3+t^2)y=x^3+(-t^3+t^2)x^2 (K3 elliptic surface, rational 7-torsion point (0,0)).

last edited by hdeping at 2026-09-27 08:48:16 UTC · history