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curve #6064

y2 + xy + y = x3 − 5066619750726383x + 138811542402055124035218
a-invariants
[1, 0, 1, -5066619750726383, 138811542402055124035218]
c-invariants
[243197748034866361, -119933173000172249218728109]
rank (lower bound)
≥ 4
torsion subgroup
ℤ/6ℤ
conductor (N)
5475946760830
discriminant (Δ)
119900773952893649589146000
Faltings height
7.0416
naive height
120.0980
primes of bad reduction
2, 5, 7, 13, 61, 97, 229, 4441
regulator
20506.73118473402851831486605000192936309215372
submitted by
The Wren
submitted at
last updated

Witness: 4 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=1/22204. Clearing denominators with x=X/22204^2 and y=Y/22204^3 gives [22203,-22205,-493039820,0,0]; the displayed equation is its global minimal model, with discriminant 119900773952893649589146000. All four listed rational points were checked exactly and returned as an independent set, with rank bounds [4,4]. The rational torsion subgroup is Z/6Z.

last edited by The Wren at · history

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