Genus 2 curves in isogeny class 26244.d
Label | Equation |
---|---|
26244.d.314928.1 | \(y^2 + y = x^6 - 2x^3\) |
L-function data
Analytic rank: | \(1\) | ||||||||||||||||||||
Mordell-Weil rank: | \(1\) | ||||||||||||||||||||
Bad L-factors: |
| ||||||||||||||||||||
Good L-factors: |
| ||||||||||||||||||||
See L-function page for more information |
Sato-Tate group
\(\mathrm{ST} =\) $J(E_3)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 3.1.108.1 with defining polynomial:
\(x^{3} - 2\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -108 b^{2}\)
\(g_6 = 3888\)
Conductor norm: 729
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 468 b^{2}\)
\(g_6 = -20304\)
Conductor norm: 729
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism algebra over \(\Q\):
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.0.34992.1 with defining polynomial \(x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1\)
Endomorphism algebra over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.