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curve #1815

y2 + xy = x3 − 114241342678577800x + 13810425722951952992443007
a-invariants
[1, 0, 0, -114241342678577800, 13810425722951952992443007]
rank (lower bound)
≥ 5
torsion subgroup
ℤ/2ℤ × ℤ/4ℤ
conductor (N)
4465971815216415
discriminant (Δ)
13027862718945297163189740607223655147886782013100625
Faltings height
8.7436
naive height
129.4449
primes of bad reduction
3, 5, 7, 17, 19, 23, 31, 37, 73, 101, 677
regulator
947.341737215444130779239251998016450213487562919
submitted by
jesper-petersen
submitted at
last updated

Witness: 5 independent points log in to add more points →

Commentary

Found by a smooth-support search in the universal family y^2=x(x+u^2)(x+v^2) with (u,v)=(18445,49932), motivated by reverse-engineering the arithmetic pattern in Eroshkin's Z/2Z x Z/4Z examples. The listed global minimal model has PARI/GP 2.17.4 rank bounds [5,5], torsion Z/2Z x Z/4Z, conductor 4465971815216415 and minimal discriminant 13027862718945297163189740607223655147886782013100625. The five listed rational points are independent; their canonical-height pairing determinant is approximately 947.3417372154. Against the snapshot of 2026-09-25, this improves the rank-5 Z/2Z x Z/4Z conductor by a factor of about 892.41 and the absolute discriminant by about 4.22e22.

last edited by jesper-petersen at · history

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