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curve #6157

y2 + xy = x3 − 16406696170231727630x + 30851590818630374211212880900
a-invariants
[1, 0, 0, -16406696170231727630, 30851590818630374211212880900]
c-invariants
[787521416171122926241, -26655774468477925442744613486961]
rank (lower bound)
≥ 4
torsion subgroup
ℤ/7ℤ
conductor (N)
324387832897590
discriminant (Δ)
-128540173864967991026733414541042638776771211920670720000000
Faltings height
10.0588
naive height
144.7211
primes of bad reduction
2, 3, 5, 7, 11, 17, 61, 135417197
regulator
704.4249089445045526693764378107691553776384073
submitted by
The Wren
submitted at
last updated

Witness: 4 independent points log in to add more points →

Commentary

Specialization at u=29/17, giving t=187/427, of Leopoldo Kulesz’s family in “Families of elliptic curves of high rank with nontrivial torsion group over Q”, Acta Arithmetica 108 (2003), Theorem 2.6, also cited on curve#1213 (David Renshaw), curve#4550 and curve#4552. https://www.impan.pl/shop/en/publication/transaction/download/product/82638 Here t=-2*(u-3)/(u^2+3), b=t^2*(t-1), c=t*(t-1), and the source equation is y^2+(1-c)*x*y-b*y=x^3-b*x^2. Selected by a bounded rational-parameter search. PARI/GP returned rank bounds [4,4], four independent points and torsion Z/7Z. All four points were checked exactly on the displayed global minimal model.

last edited by The Wren at · history

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