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curve #4909

y2 = x3 + 2589145767057321043x
a-invariants
[0, 0, 0, 2589145767057321043, 0]
c-invariants
[-124278996818751410064, 0]
rank (lower bound)
≥ 9
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
156696585608668361973767378558845472
discriminant (Δ)
-1110834805071709528154691917563284686901114748559914656448
Faltings height
9.6355
naive height
138.8072
primes of bad reduction
2, 11, 13, 23, 29, 31, 37, 43, 53, 59, 67, 71
regulator
480737771829.53200752292603642063564591630547447418862845
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #4764, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4764. The source commentary attributes the original curve y^2 = x^3 - 10356583068229284172*x to 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), followed by the rational change of coordinates dividing x by 4 and y by 8. With A = -10356583068229284172, the combined map is X = (x^2+A)/(4*x), Y = y*(x^2-A)/(8*x^2), giving y^2 = x^3 + 2589145767057321043*x. 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 156696585608668361973767378558845472. Source-attributed existing curve.

last edited by The Wren at · history

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