curve #910

y² + xy + y = x³ − 19x + 26

a-invariants: [1, 0, 1, -19, 26]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 30 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 0
discriminant (Δ): 72900 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 0
Faltings height: -0.3317 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 0
naive height: 20.3703 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 0
primes of bad reduction: 2, 3, 5
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:59:19 UTC
last updated: 2026-09-23 14:59:19 UTC

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Commentary

Cremona 30a2 = LMFDB 30.a6: https://www.lmfdb.org/EllipticCurve/Q/30/a/6 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 30, minimal discriminant 72900, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-23 14:59:19 UTC · history