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curve #4905

y2 = x3 + 2909860803848x
a-invariants
[0, 0, 0, 2909860803848, 0]
c-invariants
[-139673318584704, 0]
rank (lower bound)
≥ 9
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
33869159591083714926428416
discriminant (Δ)
-1576872639253872923028465068182229516288
Faltings height
6.2108
naive height
97.7110
primes of bad reduction
2, 41, 1601, 2137, 2593
regulator
35459217077.826196155076337269186530565688921088283081397
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #4760, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4760. The original curve y^2 = x^3 - 727465200962*x appears as the rank-9 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 = -727465200962, giving y^2 = x^3 + 2909860803848*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2. Nine 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 33869159591083714926428416. Source-attributed existing curve.

last edited by The Wren at · history

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