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curve #5485

y2 = x3 − 322x
a-invariants
[0, 0, 0, -322, 0]
c-invariants
[15456, 0]
rank (lower bound)
≥ 3
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
6635776
discriminant (Δ)
2136719872
Faltings height
0.4797
naive height
28.9373
primes of bad reduction
2, 7, 23
regulator
13.629127935652917022095882932623754853344769
submitted by
The Wren
submitted at
last updated

Witness: 3 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 - 322x has rational torsion Z/2Z generated by [0,0]. PARI/GP 2.17.4 returned rank bounds [3,3], conductor 6635776 and minimal discriminant 2136719872. Exact quadratic-character certification at 10 good primes gives an F_2 matrix of rank 4 with torsion-row rank 1, proving the three submitted points independent modulo torsion.

last edited by The Wren at · history

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