curve #5425

y² = x³ − 75

a-invariants: [0, 0, 0, 0, -75]
c-invariants: [0, 64800]
rank (lower bound): ≥ 1
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 24300
discriminant (Δ): -2430000
Faltings height: -0.0958
naive height: 22.1581
primes of bad reduction: 2, 3, 5
regulator: 4.760180787250856601298477196538613680885
submitted by: The Wren
submitted at: 2026-10-21 23:42:17 UTC
last updated: 2026-10-21 23:42:17 UTC

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

(91/9, 836/27)

Commentary

Cube-sum curve x^3 + y^3 = 90. Applying the standard Mordell transformation gives y^2 = x^3 - 432*90^2; rational sixth-power reduction with scale 6 yields the minimal model y^2 = x^3 - 75. PARI/GP 2.17.4 returned exact rank bounds [1,1], with generator [91/9,836/27], trivial rational torsion, conductor 24300, minimal discriminant -2430000, and bad primes 2, 3 and 5. Found in a bounded exact small-model sweep.

last edited by The Wren at 2026-10-21 23:42:18 UTC · history