Genus 2 isogeny class 147456.e

• Label: 147456.e
• Conductor: $147456$
• Sato-Tate group: $J(E_4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 147456.e

Label	Equation
147456.e.884736.1	\(y^2 = x^5 - 5x^3 + 6x\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(3\)	\( 1 + 3 T^{2}\)

Good L-factors:

Prime	L-Factor
\(5\)	\( ( 1 + 5 T^{2} )^{2}\)
\(7\)	\( 1 + 6 T^{2} + 49 T^{4}\)
\(11\)	\( 1 - 10 T^{2} + 121 T^{4}\)
\(13\)	\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)	\( 1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)	\( ( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)	\( ( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)	\( 1 - 54 T^{2} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_4)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.2.55296.1 with defining polynomial:
  \(x^{4} - 6\)

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
  \(g_4 = 960 b^{2} + 1440\)
  \(g_6 = -34560 b^{3} - 64512 b\)
   Conductor norm: 32
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 960 b^{2} + 1440\)
  \(g_6 = 34560 b^{3} + 64512 b\)
   Conductor norm: 32

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 \(K = \Q (a) \simeq \) 8.0.12230590464.4 with defining polynomial \(x^{8} + 4 x^{6} - 6 x^{4} + 76 x^{2} + 25\)

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.