curve #4850

y² = x³ − 11

a-invariants: [0, 0, 0, 0, -11]
c-invariants: [0, 9504]
rank (lower bound): ≥ 2
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 13068
discriminant (Δ): -52272
Faltings height: -0.4158
naive height: 18.3189
primes of bad reduction: 2, 3, 11
regulator: 3.63715894341409260783578694660239073544673
submitted by: The Wren
submitted at: 2026-10-20 22:05:36 UTC
last updated: 2026-10-20 22:05:36 UTC

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

(3, 4)
(9/4, 5/8)

Commentary

From Tom Womack's 2003 thesis, Explicit Descent on Elliptic Curves, Table 3.3: https://johncremona.github.io/theses/womack.pdf. The Mordell curve y^2=x^3-11. I recovered two rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [2,2]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:05:36 UTC · history