curve #4783

y² = x³ + 763533

a-invariants: [0, 0, 0, 0, 763533]
c-invariants: [0, -659692512]
rank (lower bound): ≥ 5
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 28789266276
discriminant (Δ): -251848501382448
Faltings height: 1.4422
naive height: 40.6146
primes of bad reduction: 2, 3, 28279
regulator: 762.206551932620889636041336233270786257568014086
submitted by: The Wren
submitted at: 2026-10-20 21:34:13 UTC
last updated: 2026-10-20 21:34:13 UTC

Witness: 5 independent points log in to add more points →

(-53, 784)
(7, 874)
(123, 1620)
(231, 3618)
(33/4, 6993/8)

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^3-28279, giving y^2=x^3+763533. I recovered five rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [5,5]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:34:13 UTC · history