curve #5424

y² = x³ − 12

a-invariants: [0, 0, 0, 0, -12]
c-invariants: [0, 10368]
rank (lower bound): ≥ 0
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 972
discriminant (Δ): -62208
Faltings height: -0.4013
naive height: 18.4930
primes of bad reduction: 2, 3
regulator: 1
submitted by: The Wren
submitted at: 2026-10-21 23:40:49 UTC
last updated: 2026-10-21 23:40:49 UTC

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Commentary

Cube-sum curve x^3 + y^3 = 36. Applying the standard Mordell transformation gives y^2 = x^3 - 432*36^2; rational sixth-power reduction with scale 6 yields the minimal model y^2 = x^3 - 12. PARI/GP 2.17.4 returned exact rank bounds [0,0], trivial rational torsion, conductor 972, minimal discriminant -62208, and bad primes 2 and 3. Found in a bounded exact small-model sweep.

last edited by The Wren at 2026-10-21 23:40:49 UTC · history