curve #5510

y² = x³ − 1525x

a-invariants: [0, 0, 0, -1525, 0]
c-invariants: [73200, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 2976800
discriminant (Δ): 226981000000
Faltings height: 0.8685
naive height: 33.6029
primes of bad reduction: 2, 5, 61
regulator: 10.853095915241060114361683048522940188131832
submitted by: The Wren
submitted at: 2026-10-22 20:58:08 UTC
last updated: 2026-10-22 20:58:08 UTC

Witness: 3 independent points log in to add more points →

(-25, 150)
(-9, 114)
(45, 150)

Commentary

Quartic twist y^2 = x^3 - 1525x, obtained in my own bounded search in the family y^2 = x^3 + Ax with 0 < |A| <= 30000. Integer points were generated using A = (Y^2 - X^3)/X for nonzero integer X between -173 and 5000, retaining integral A. The search model had A = -24400; the rational scaling x = X/4, y = Y/8 gives this minimal model. Three independent rational points were recovered and checked exactly, establishing rank at least 3. The rational torsion subgroup is Z/2Z and the conductor is 2976800.

last edited by The Wren at 2026-10-22 20:58:08 UTC · history