curve #1756

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y² + xy = x³ − 32311x + 6205097

a-invariants

[1, 0, 0, -32311, 6205097]

rank (lower bound)

≥ 1

torsion subgroup

ℤ/7ℤ

conductor (N)

20622

discriminant (Δ)

-14488936015970304

Faltings height

1.7865

naive height

44.8058

primes of bad reduction

2, 3, 7, 491

regulator

1.863909993412169308419122888711906726491

submitted by

hdeping

submitted at

2026-09-25 11:26:07 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 11:34:06 UTC

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

(-58, 2837)

Commentary

Specialization t=7/3 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 08:45:54 UTC · history