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curve #1190

y2 + xy = x3 − 33089897206260x + 72431786525672019600
a-invariants
[1, 0, 0, -33089897206260, 72431786525672019600]
rank (lower bound)
≥ 2
torsion subgroup
ℤ/2ℤ × ℤ/8ℤ
conductor (N)
169350090 ☆ record for ℤ/2ℤ × ℤ/8ℤ torsion, rank ≥ 2
discriminant (Δ)
52386387326982250326891111127449600000000 ☆ record for ℤ/2ℤ × ℤ/8ℤ torsion, rank ≥ 2
Faltings height
6.6211 ☆ record for ℤ/2ℤ × ℤ/8ℤ torsion, rank ≥ 2
naive height
105.0044 ☆ record for ℤ/2ℤ × ℤ/8ℤ torsion, rank ≥ 2
primes of bad reduction
2, 3, 5, 7, 13, 17, 41, 89
regulator
16.1451419529730691095098975188826762911041
submitted by
David Renshaw
submitted at
last updated

Witness: 2 independent points log in to add more points →

Commentary

Found by a rational-parameter search of y^2=x(x+u^4)(x+v^4), with u=m^2-n^2 and v=2mn, at (m,n)=(8,5). Converted to the global minimal model; PARI/GP ellrank returned bounds [2,2]. The 2 supplied points and the torsion subgroup are independently certified by the site.

last edited by David Renshaw at · history

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