curve #1748

y² + xy = x³ − 319040x + 74649600

a-invariants: [1, 0, 0, -319040, 74649600]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/7ℤ
conductor (N): 27690
discriminant (Δ): -330727587840000000
Faltings height: 2.1043
naive height: 49.7805
primes of bad reduction: 2, 3, 5, 13, 71
regulator: 1.751933353676479625160961243000059873368
submitted by: hdeping
submitted at: 2026-09-25 10:47:35 UTC
last updated: 2026-09-25 10:47:35 UTC

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

(-620, 6160)

Commentary

Specialization t=-5/3 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 08:46:58 UTC · history