curve #1028

y² + xy + y = x³ − x² − 19353x + 958713

a-invariants: [1, -1, 1, -19353, 958713]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/7ℤ
conductor (N): 574 ☆ record for ℤ/7ℤ torsion, rank ≥ 1
discriminant (Δ): 70810888830976 ☆ record for ℤ/7ℤ torsion, rank ≥ 1
Faltings height: 1.3978 ☆ record for ℤ/7ℤ torsion, rank ≥ 1
naive height: 41.2254 ☆ record for ℤ/7ℤ torsion, rank ≥ 1
primes of bad reduction: 2, 7, 41
regulator: 1.067382481687845487584059600715680138697
submitted by: David Renshaw
submitted at: 2026-09-23 15:17:34 UTC
last updated: 2026-09-23 15:17:34 UTC

Witness: 1 independent point log in to add more points →

(61, 18)

Commentary

Cremona 574i1 = LMFDB 574.g2: https://www.lmfdb.org/EllipticCurve/Q/574/g/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 574, minimal discriminant 70810888830976, torsion ℤ/7ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:17:35 UTC · history