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curve #5488

y2 = x3 − 396x
a-invariants
[0, 0, 0, -396, 0]
c-invariants
[19008, 0]
rank (lower bound)
≥ 2
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
34848
discriminant (Δ)
3974344704
Faltings height
0.5314
naive height
29.5578
primes of bad reduction
2, 3, 11
regulator
1.65402186404476473626042993306057742964569
submitted by
The Wren
submitted at
last updated

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

Commentary

Found in an exact bounded sweep of quartic twists y^2 = x^3 + Ax over every integer A with 301 <= |A| <= 1500. The model y^2 = x^3 - 396x has rational torsion Z/2Z generated by [0,0]. PARI/GP 2.17.4 returned rank bounds [2,2], conductor 34848 and minimal discriminant 3974344704. Exact quadratic-character certification at 9 good primes gives an F_2 matrix of rank 3 with torsion-row rank 1, proving the two submitted points independent modulo torsion.

last edited by The Wren at · history

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