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curve #4916

y2 = x3 − 5169170820204434510666892x
a-invariants
[0, 0, 0, -5169170820204434510666892, 0]
c-invariants
[248120199369812856512010816, 0]
rank (lower bound)
≥ 12
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
5937850437433996936516077156029887017540129986592
discriminant (Δ)
8839803806185522806373912781989731104006643263104402253996144963598233874432
Faltings height
13.2622
naive height
182.3279
primes of bad reduction
2, 3, 7, 11, 19, 23, 47, 67, 103, 127, 149, 353, 733, 2687
regulator
8043198251240090.6254460042389333823161456785097603217441708100
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #4768, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4768. The source commentary identifies #4768 as a 2-isogenous model of a curve listed in Aguirre, Castañeda and Peral, High rank elliptic curves with torsion group Z/(2Z), Mathematics of Computation 73 (2004), 323–331: https://doi.org/10.1090/S0025-5718-03-01547-3. Applied the standard degree-2 isogeny with kernel generated by (0,0), sending y^2 = x^3 + A*x to y^2 = x^3 - 4*A*x. Here A = 1292292705051108627666723, giving y^2 = x^3 - 5169170820204434510666892*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2. Twelve supplied rational points transported through this map and checked exactly on the target curve. Rational torsion C2. Same rank and conductor as the source; conductor 5937850437433996936516077156029887017540129986592. Source-attributed existing curve.

last edited by The Wren at · history

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