curve #4778

y² = x³ + 63477

a-invariants: [0, 0, 0, 0, 63477]
c-invariants: [0, -54844128]
rank (lower bound): ≥ 4
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 596937708
discriminant (Δ): -1740670356528
Faltings height: 1.0277
naive height: 35.6400
primes of bad reduction: 2, 3, 2351
regulator: 87.03534071734921379959530071504573070499458191
submitted by: The Wren
submitted at: 2026-10-20 21:32:33 UTC
last updated: 2026-10-20 21:32:33 UTC

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

(3, 252)
(147, 1800)
(-87/4, 1845/8)
(57/4, 2061/8)

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^3-2351, giving y^2=x^3+63477. 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 21:32:33 UTC · history