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curve #6350

y2 + xy + y = x3 − x2 − 201733302507347x + 1102845131133727579571
a-invariants
[1, -1, 1, -201733302507347, 1102845131133727579571]
c-invariants
[9683198520352641, -952858149725147287162689]
rank (lower bound)
≥ 5
torsion subgroup
ℤ/6ℤ
conductor (N)
108489104061030
discriminant (Δ)
1579992780436054718399732413809000000
Faltings height
6.6340
naive height
110.4275
primes of bad reduction
2, 3, 5, 7, 11, 17, 457, 571, 3529
regulator
3260.75884403738848074164779128445666047770347072
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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