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curve #1763

y2 = x3 − x2 − 10305216940672333963680x + 398834982658354598384375680358400
a-invariants
[0, -1, 0, -10305216940672333963680, 398834982658354598384375680358400]
rank (lower bound)
≥ 5
torsion subgroup
ℤ/2ℤ × ℤ/4ℤ
conductor (N)
140233810193909992560
discriminant (Δ)
1322890813477155550797401760434272274862687898175567556000000000000
Faltings height
11.5024
naive height
163.6744
primes of bad reduction
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 43, 53, 79, 101, 149
regulator
2120.60868924969973469840340741125353597634593340
submitted by
jesper-petersen
submitted at
last updated

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Commentary

Specialization t=-3/2 of the generic rank-4 family with torsion Z/2Z x Z/4Z constructed by Dujella and Peral (LMS J. Comput. Math. 17 (2014), 282-288). The listed global minimal model has PARI/GP 2.17.4 rank bounds [5,5], torsion Z/2Z x Z/4Z, conductor 140233810193909992560 and minimal discriminant 1322890813477155550797401760434272274862687898175567556000000000000. The five listed rational points are independent; their canonical-height pairing determinant is approximately 2120.6086892497. Against the database snapshot of 2026-09-25, this improves the absolute discriminant of the existing rank-5 Z/2Z x Z/4Z entry #1376 by a factor of approximately 4.153307779e8.

last edited by jesper-petersen at · history

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