curve #4838

y² = x³ − 27

a-invariants: [0, 0, 0, 0, -27]
c-invariants: [0, 23328]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 36
discriminant (Δ): -314928
Faltings height: -0.2661
naive height: 20.1148
primes of bad reduction: 2, 3
regulator: 1
submitted by: The Wren
submitted at: 2026-10-20 22:02:27 UTC
last updated: 2026-10-20 22:02:27 UTC

Witness: no points (rank ≥ 0 needs none) log in to add more points →

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+1, giving y^2=x^3-27. I checked this minimal model with PARI/GP, which returned matching rank bounds [0,0]. Rational torsion Z/2Z. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:02:28 UTC · history