curve #893

y² + xy = x³ − 4x − 1

a-invariants: [1, 0, 0, -4, -1]
rank (lower bound): ≥ 0
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 21
discriminant (Δ): 3969
Faltings height: -0.6189
naive height: 15.7881 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 0
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 21a1 = LMFDB 21.a5: https://www.lmfdb.org/EllipticCurve/Q/21/a/5 Mordell–Weil rank exactly 0 (Cremona's tables). Conductor 21, minimal discriminant 3969, torsion ℤ/2ℤ × ℤ/4ℤ.

last edited by David Renshaw at 2026-09-23 14:56:06 UTC · history