curve #896

y² + xy = x³ + x

a-invariants: [1, 0, 0, 1, 0]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/4ℤ
conductor (N): 21
discriminant (Δ): -63
Faltings height: -0.9655
naive height: 11.5504
primes of bad reduction: 3, 7
regulator: 1
submitted by: David Renshaw
submitted at: 2026-09-23 14:57:09 UTC
last updated: 2026-09-23 14:57:09 UTC

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Commentary

Cremona 21a4 = LMFDB 21.a6: https://www.lmfdb.org/EllipticCurve/Q/21/a/6 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 21, minimal discriminant -63, torsion ℤ/4ℤ.

last edited by David Renshaw at 2026-09-23 14:57:09 UTC · history