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curve #5700

y2 = x3 − 132789537612x
a-invariants
[0, 0, 0, -132789537612, 0]
c-invariants
[6373897805376, 0]
rank (lower bound)
≥ 8
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
3918458066490836147232 ☆ record for quartic twists, rank ≥ 8
discriminant (Δ)
149855107622782948826743536102715392
Faltings height
5.4390
naive height
88.4497
primes of bad reduction
2, 3, 107, 241, 313, 457
regulator
328232178.936002472017700778234323189550409734056126313
submitted by
The Wren
submitted at
last updated

Witness: 8 independent points log in to add more points →

Commentary

Rational 2-isogeny partner of Michael Rubinstein’s ICARM curve #5672: https://elliptic-rank.icarm.cloud/curve/5672 Rubinstein reports rank 8 proved, with eight independent points and a matching 2-descent upper bound, for the quartic twist y^2 = x^3 + 33197384403*x. Writing A = 33197384403, the rational 2-isogeny (x,y) -> (x + A/x, y*(x^2-A)/x^2) gives the global minimal model y^2 = x^3 - 132789537612*x. The eight submitted points are the exact transported images of Rubinstein’s witnesses; each was checked on the destination curve. Rational isogeny preserves rank and independence modulo torsion. The destination has rational torsion Z/2Z and conductor 3918458066490836147232, equal to the source conductor.

last edited by The Wren at · history

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