curve #1185

y² + xy + y = x³ + x² − 270010x + 59128551

a-invariants: [1, 1, 1, -270010, 59128551]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/5ℤ
conductor (N): 3559178
discriminant (Δ): -256252875388461728 ☆ record for ℤ/5ℤ torsion, rank ≥ 3
Faltings height: 2.0763 ☆ record for ℤ/5ℤ torsion, rank ≥ 3
naive height: 49.3174 ☆ record for ℤ/5ℤ torsion, rank ≥ 3
primes of bad reduction: 2, 7, 37, 6871
regulator: 25.328383630871813151698326778280537918378323
submitted by: David Renshaw
submitted at: 2026-09-23 21:39:57 UTC
last updated: 2026-09-23 21:39:57 UTC

Witness: 3 independent points log in to add more points →

(-499, 8611)
(-1293/4, 86241/8)
(-277, 10757)

Commentary

Found by a rational-parameter search of the Kubert-Tate family for torsion 5, at t=-37/14. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t;b=t; Converted to the global minimal model; PARI/GP ellrank returned bounds [3,3]. The 3 supplied points and the torsion subgroup are independently certified by the site.

last edited by David Renshaw at 2026-09-23 21:39:57 UTC · history