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curve #806

y2 = x3 + x2 − 48644x − 2108730
a-invariants
[0, 1, 0, -48644, -2108730]
rank (lower bound)
≥ 5
torsion subgroup
ℤ/2ℤ
conductor (N)
9353416128
discriminant (Δ)
5475199845431232
Faltings height
1.7163
naive height
43.9905
primes of bad reduction
2, 3, 7, 107, 193, 337
regulator
584.393501974103804655999204205864744894449557682
submitted by
benvchurch
submitted at
last updated

Witness: 5 independent points log in to add more points →

Commentary

Found on 2026-09-19 by searching a second elliptic fibration of the D=26 K3 companion at moduli parameter j=-2. The source equation is Y^2=X^3+t^3(72900-3267t)X-t^5(43040889+405756t+729t^2). Setting u=X/t^2 gives a rational-2-torsion pencil; this curve is its fiber u=-60 (scaled parameter v=20). Candidates were selected by a Mestre-Nagao prime-sum sieve through 251 on a reduced-rational box |numerator(v)|<=256, denominator(v)<=8, followed by Magma descent and rational-point computations. In a fresh process with rigorous Minkowski class-group bounds, Magma verified the five submitted points, their independence modulo torsion, and unconditional rank bounds [5,5]. Thus the rank is exactly 5. Magma also verified global minimality, torsion Z/2, and conductor 9353416128. The exact map back to the source K3 is t=-(x+195), X=-60t^2, Y=27t^2y. The submitted points are independent witnesses; they are not claimed to be a saturated basis.

last edited by benvchurch at · history

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