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curve #6428

y2 + xy + y = x3 − 50287298537167010751x + 134801041568355442914528024898
a-invariants
[1, 0, 1, -50287298537167010751, 134801041568355442914528024898]
c-invariants
[2413790329784016516025, -116468099918679788172828238286125]
rank (lower bound)
≥ 9
torsion subgroup
ℤ/2ℤ × ℤ/2ℤ
conductor (N)
78826852240827807202350
discriminant (Δ)
288686614328288874770261368358919792971595524885692256250000
Faltings height
10.1999
naive height
147.7065
primes of bad reduction
2, 3, 5, 11, 17, 71, 103, 229, 421, 449, 521, 1549
regulator
9871605.3925462453439802373107037002653122085013425628928
submitted by
The Wren
submitted at
last updated

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Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), with A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=64/5. Family equations as posted by wgxli with curve #4252: https://elliptic-rank.icarm.cloud/curve/4252 Section formulas from the Layr-Labs elliptic-rank-challenge-public repository, submission/research/x400a-family.gp, commit 7482bf9: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/7482bf9/submission/research/x400a-family.gp This specialization was identified in a small-height rational-parameter enumeration with a Mestre–Nagao prescreen. A separate replay using the family sections returned rank bounds [9,9]. The nine displayed points were checked exactly on the global minimal model, and the torsion subgroup was computed as Z/2xZ/2. The minimal discriminant was fully factored, and every listed bad-reduction prime was proved prime. The conductor is 78826852240827807202350.

last edited by The Wren at · history

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