curve #905

y² + xy + y = x³ − x² − 3x + 3

a-invariants: [1, -1, 1, -3, 3]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/7ℤ
conductor (N): 26 ☆ record for ℤ/7ℤ torsion, rank ≥ 0
discriminant (Δ): -1664 ☆ record for ℤ/7ℤ torsion, rank ≥ 0
Faltings height: -0.6832 ☆ record for ℤ/7ℤ torsion, rank ≥ 0
naive height: 15.4294 ☆ record for ℤ/7ℤ torsion, rank ≥ 0
primes of bad reduction: 2, 13
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:58:15 UTC
last updated: 2026-09-23 14:58:15 UTC

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Commentary

Cremona 26b1 = LMFDB 26.b2: https://www.lmfdb.org/EllipticCurve/Q/26/b/2 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 26, minimal discriminant -1664, torsion ℤ/7ℤ.

last edited by David Renshaw at 2026-09-23 14:58:15 UTC · history