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curve #4904

y2 = x3 + 565049243588x
a-invariants
[0, 0, 0, 565049243588, 0]
c-invariants
[-27122363692224, 0]
rank (lower bound)
≥ 8
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
638561295358741918227488
discriminant (Δ)
-11546194461664978431928007811499102208
Faltings height
5.8011
naive height
92.7941
primes of bad reduction
2, 41, 769, 2081, 2153
regulator
2294771375.45046611941559249506367199470323422854298896
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #4759, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4759. The original curve y^2 = x^3 - 141262310897*x appears as the rank-8 example in Table 2, Section 6.1, of Julián Aguirre, Álvaro Lozano-Robledo and Juan Carlos Peral, Elliptic curves of maximal rank: https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf. 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 = -141262310897, giving y^2 = x^3 + 565049243588*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2. Eight 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 638561295358741918227488. Source-attributed existing curve.

last edited by The Wren at · history

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