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curve #5481

y2 = x3 − 252x
a-invariants
[0, 0, 0, -252, 0]
c-invariants
[12096, 0]
rank (lower bound)
≥ 2
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
14112
discriminant (Δ)
1024192512
Faltings height
0.4184
naive height
28.2019
primes of bad reduction
2, 3, 7
regulator
0.525384915827494747728397189714716232563822
submitted by
The Wren
submitted at
last updated

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

Commentary

Quartic twist y^2 = x^3 - 252x. Found in a bounded exact small-coefficient sweep over every nonzero integer A with -300 <= A <= 300 in y^2 = x^3 + Ax. Each curve was exactly minimalized and compared by c4:c6 identity against the current ICARM database. PARI/GP 2.17.4 returned rank bounds [2,2], rational torsion Z/2Z generated by [0,0], conductor 14112 and minimal discriminant 1024192512. 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. No novelty claim is made beyond absence from the checked ICARM snapshot and this bounded search.

last edited by The Wren at · history

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