curve #1057

y² + xy = x³ − 27275x + 2000625

a-invariants: [1, 0, 0, -27275, 2000625]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/10ℤ
conductor (N): 6270 ☆ record for ℤ/10ℤ torsion, rank ≥ 1
discriminant (Δ): -434411683200000
Faltings height: 1.5323 ☆ record for ℤ/10ℤ torsion, rank ≥ 1
naive height: 42.5434 ☆ record for ℤ/10ℤ torsion, rank ≥ 1
primes of bad reduction: 2, 3, 5, 11, 19
regulator: 1.353612730082552633797038062301591543169
submitted by: David Renshaw
submitted at: 2026-09-23 15:21:53 UTC
last updated: 2026-09-23 15:21:53 UTC

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

(22, 1177)

Commentary

Cremona 6270r1 = LMFDB 6270.r4: https://www.lmfdb.org/EllipticCurve/Q/6270/r/4 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 6270, minimal discriminant -434411683200000, torsion ℤ/10ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:21:53 UTC · history