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curve #6351

y2 + xy + y = x3 − 12299514469658x + 16602528070308124568
a-invariants
[1, 0, 1, -12299514469658, 16602528070308124568]
c-invariants
[590376694543561, -14344585138311261442309]
rank (lower bound)
≥ 5
torsion subgroup
ℤ/6ℤ
conductor (N)
32188214579430
discriminant (Δ)
3188967518535680397712318647095434500
Faltings height
6.1393
naive height
102.0353
primes of bad reduction
2, 3, 5, 7, 13, 23, 37, 79, 83, 2113
regulator
11422.4542415980019144401316842739924037325539408
submitted by
The Wren
submitted at
last updated

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Commentary

Z/6-torsion Kubert–Tate family curve from a rational-parameter search of the family y^2+(1-c)xy-by=x^3-bx^2 with c=a/d, b=c(c+1), following the construction of David Renshaw on curve #1192 and the t=1/6916 example of Lewis Combes on curve #5970. The search covered coprime pairs (a,d) with log|Δ_min| ≤ 88.12, ranked by discriminant. Rank ≥ 5 certified: 5 independent rational points, checked exactly (independent mod 5 and mod 7); PARI ellrank bounds 5 5. The rational torsion subgroup is Z/6Z.

last edited by The Wren at · history

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