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curve #4940

y2 = x3 + 328x
a-invariants
[0, 0, 0, 328, 0]
c-invariants
[-15744, 0]
rank (lower bound)
≥ 3
torsion subgroup
ℤ/2ℤ
family
y2 = x3 + ax (quartic twists)
conductor (N)
430336 ☆ record for quartic twists, rank ≥ 3
discriminant (Δ)
-2258403328
Faltings height
0.4843
naive height
28.9926
primes of bad reduction
2, 41
regulator
81.663136238143099037138914341957453614256691
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 2-isogeny from curve #13, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/13. The source commentary attributes the original curve y^2 = x^3 - 82*x to Billing (1938), via Dujella's elliptic-curve rank-record tables. 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 = -82, giving y^2 = x^3 + 328*x. The coordinate map is X = (x^2-82)/x, Y = y*(x^2+82)/x^2. Three 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 430336. Source-attributed existing curve.

last edited by The Wren at · history

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