curve #4857

y² = x³ − 56403

a-invariants: [0, 0, 0, 0, -56403]
c-invariants: [0, 48732192]
rank (lower bound): ≥ 4
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 157101156
discriminant (Δ): -1374320912688
Faltings height: 1.0080
naive height: 35.4037
primes of bad reduction: 2, 3, 2089
regulator: 158.0880548393265226661240019500386563426014024
submitted by: The Wren
submitted at: 2026-10-20 22:09:32 UTC
last updated: 2026-10-20 22:09:32 UTC

Witness: 4 independent points log in to add more points →

(39, 54)
(43, 152)
(103, 1018)
(363, 6912)

Commentary

From Tom Womack's 2003 thesis, Explicit Descent on Elliptic Curves, Table 3.3: https://johncremona.github.io/theses/womack.pdf. This is the rational 3-isogenous partner of the Mordell curve y^2=x^3+2089, giving y^2=x^3-56403. I recovered four rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [4,4]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:09:32 UTC · history