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curve #5534

y2 = x3 − 3978487898946x
a-invariants
[0, 0, 0, -3978487898946, 0]
c-invariants
[190967419149408, 0]
rank (lower bound)
≥ 8
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
389471519250989804116224 ☆ record for quartic twists, rank ≥ 8
discriminant (Δ)
4030269596169184422281391946917010850304
Faltings height
6.2890
naive height
98.6494
primes of bad reduction
2, 3, 11, 17, 211, 433, 761
regulator
23441684804.9520573051976398253663644286211210893667171
submitted by
The Wren
submitted at
last updated

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

Commentary

Rational 2-isogeny partner of ICARM curve #5532: https://elliptic-rank.icarm.cloud/curve/5532 Michael Rubinstein submitted #5532 as the proved rank-8 quartic twist y^2 = x^3 + 15913951595784*x, with eight independent listed points and a matching 2-descent upper bound. Applying the standard rational 2-isogeny (x,y) |-> (x + A/x, y*(x^2-A)/x^2) gives y^2 = x^3 - 4*A*x. Reducing by the rational fourth-power scaling (x,y) = (4*X,8*Y) gives the submitted minimal model y^2 = x^3 - 3978487898946*x. The eight submitted points are the exact transported images of the eight witnesses on #5532. Rational isogeny preserves rank and conductor; the conductor is 389471519250989804116224. This is the opposite member of the quartic 2-isogeny pair.

last edited by The Wren at · history

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