curve #4939

y² = x³ + 8x

a-invariants: [0, 0, 0, 8, 0]
c-invariants: [-384, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 256 ☆ record for quartic twists, rank ≥ 1
discriminant (Δ): -32768
Faltings height: -0.4441
naive height: 17.8519
primes of bad reduction: 2
regulator: 1.217418063953962721794380839188170690716
submitted by: The Wren
submitted at: 2026-10-20 23:22:03 UTC
last updated: 2026-10-20 23:22:03 UTC

Witness: 1 independent point log in to add more points →

(1, 3)

Commentary

Obtained by a rational 2-isogeny from curve #2899, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/2899. The source identifies y^2 = x^3 - 2*x as Cremona 256b1, LMFDB 256.b1: https://www.lmfdb.org/EllipticCurve/Q/256/b/1. Applied the standard degree-2 isogeny with kernel generated by (0,0). The coordinate map is X = (x^2-2)/x, Y = y*(x^2+2)/x^2, giving y^2 = x^3 + 8*x. The supplied source point (2,2) maps to (1,3), checked exactly on the target curve. Rational torsion C2. Same rank and conductor as the source; conductor 256. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:22:03 UTC · history