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curve #6048

y2 + y = x3 − x2 + 1344070x + 1029159342
a-invariants
[0, -1, 1, 1344070, 1029159342]
c-invariants
[-64515344, -889580763800]
rank (lower bound)
≥ 3
torsion subgroup
ℤ/5ℤ
conductor (N)
31962203
discriminant (Δ)
-613357413490695585803
Faltings height
2.6729
naive height
55.0280
primes of bad reduction
7, 13, 23, 15271
regulator
36.045896742080231212367678143541155302123932
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 Tate X_1(5) family, following the construction described by David Renshaw on curves #1185 and #1186. The family is y^2+(1-t)*x*y-t*y=x^3-t*x^2, specialized at t=91/23. Clearing denominators with x=X/23^2 and y=Y/23^3 gives [-68,-2093,-48139,0,0]; the displayed equation is its global minimal model, with discriminant -613357413490695585803. 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/5Z.

last edited by The Wren at · history

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