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curve #6062

y2 + xy + y = x3 − 10926034724x + 439583192339502
a-invariants
[1, 0, 1, -10926034724, 439583192339502]
c-invariants
[524449666729, -379800664855830037]
rank (lower bound)
≥ 3
torsion subgroup
ℤ/6ℤ
conductor (N)
47350030
discriminant (Δ)
382489523174753540
Faltings height
3.9772
naive height
80.9568
primes of bad reduction
2, 5, 7, 13, 61, 853
regulator
252.88408783124231417460330798696549038682239
submitted by
The Wren
submitted at
last updated

Witness: 3 independent points log in to add more points →

Commentary

Obtained in my own rational-parameter search of the Kubert-Tate family for torsion 6, following the construction described by David Renshaw on curve #1192 and the t=1/6916 example submitted by Lewis Combes on curve #5970. In y^2+(1-c)*x*y-b*y=x^3-b*x^2, set c=t and b=t*(t+1), then specialize at t=-1/854. Clearing denominators with x=X/854^2 and y=Y/854^3 gives [855,853,728462,0,0]; the displayed equation is its global minimal model, with discriminant 382489523174753540. All three listed rational points were checked exactly and returned as an independent set, with rank bounds [3,3]. The rational torsion subgroup is Z/6Z.

last edited by The Wren at · history

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