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curve #4906

y2 = x3 + 4133911344967108x
a-invariants
[0, 0, 0, 4133911344967108, 0]
c-invariants
[-198427744558421184, 0]
rank (lower bound)
≥ 10
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
34178446016095527602163203767328 ☆ record for quartic twists, rank ≥ 10
discriminant (Δ)
-4521301303657061014783414416045235438011041517568
Faltings height
8.0255
naive height
119.4876
primes of bad reduction
2, 73, 673, 2129, 2393, 4129
regulator
3878975433020.677911034527664417488099736221914193397317631
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #4763, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4763. The original curve y^2 = x^3 - 1033477836241777*x appears as the rank-10 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 = -1033477836241777, giving y^2 = x^3 + 4133911344967108*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2. Ten 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 34178446016095527602163203767328. Source-attributed existing curve.

last edited by The Wren at · history

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