curve #894

y² + xy = x³ − 49x − 136

a-invariants: [1, 0, 0, -49, -136]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 21
discriminant (Δ): 21609
Faltings height: -0.2724
naive height: 23.2903
primes of bad reduction: 3, 7
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:56:06 UTC
last updated: 2026-09-23 14:56:06 UTC

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Commentary

Cremona 21a2 = LMFDB 21.a2: https://www.lmfdb.org/EllipticCurve/Q/21/a/2 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 21, minimal discriminant 21609, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-23 14:56:07 UTC · history