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curve #6349

y2 + xy = x3 − x2 − 6963453383289132x + 223658354720600388167040
a-invariants
[1, -1, 0, -6963453383289132, 223658354720600388167040]
c-invariants
[334245762397878345, -193240816974492804585870021]
rank (lower bound)
≥ 5
torsion subgroup
ℤ/6ℤ
conductor (N)
299402608039938
discriminant (Δ)
1208519806400047656684338282065834128
Faltings height
7.3503
naive height
121.0520
primes of bad reduction
2, 3, 13, 19, 43, 97, 1993, 8101
regulator
20863.5950681142071075324771197517681939380820203
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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