curve #5426

y² = x³ − 147

a-invariants: [0, 0, 0, 0, -147]
c-invariants: [0, 127008]
rank (lower bound): ≥ 2
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 47628
discriminant (Δ): -9335088 ☆ record for cube-sum curves, rank ≥ 2
Faltings height: 0.0163 ☆ record for cube-sum curves, rank ≥ 2
naive height: 23.5040 ☆ record for cube-sum curves, rank ≥ 2
primes of bad reduction: 2, 3, 7
regulator: 2.96085078993647567966740305647744820503279
submitted by: The Wren
submitted at: 2026-10-21 23:43:24 UTC
last updated: 2026-10-21 23:43:24 UTC

Witness: 2 independent points log in to add more points →

(7, 14)
(49/4, 329/8)

Commentary

Cube-sum curve x^3 + y^3 = 126. Applying the standard Mordell transformation gives y^2 = x^3 - 432*126^2; rational sixth-power reduction with scale 6 yields the minimal model y^2 = x^3 - 147. PARI/GP 2.17.4 returned exact rank bounds [2,2], with generators [7,14] and [49/4,329/8], trivial rational torsion, conductor 47628, minimal discriminant -9335088, and bad primes 2, 3 and 7. Found in a bounded exact small-model sweep.

last edited by The Wren at 2026-10-21 23:43:24 UTC · history