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curve #4951

y2 = x3 + 994485754408x
a-invariants
[0, 0, 0, 994485754408, 0]
c-invariants
[-47735316211584, 0]
rank (lower bound)
≥ 9
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
3956007662881795565721856 ☆ record for quartic twists, rank ≥ 9
discriminant (Δ)
-62947092241037302440104465748063059968
Faltings height
5.9424
naive height
94.4901
primes of bad reduction
2, 17, 101, 229, 409, 773
regulator
282749165670.81583441832620789200777929161702412186176046
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #4690, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4690. The source model is y^2 = x^3 - 248621438602*x. Renshaw's commentary reports finding it using a Mestre-Nagao sieve over |a| <= 10^12, with a_p computed via quartic residue classes for primes p = 1 mod 4, followed by PARI/GP ellrank certification. Applied the degree-2 isogeny X = (x^2 + A)/x, Y = y*(x^2 - A)/x^2 with A = -248621438602. The resulting model is Y^2 = X^3 + 994485754408*X. Nine supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 3956007662881795565721856. Source-attributed existing curve.

last edited by The Wren at · history

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