Genus 2 isogeny class 43264.c

• Label: 43264.c
• Conductor: $43264$
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 43264.c

Label	Equation
43264.c.43264.1	\(y^2 = x^5 - 5x^3 + 5x^2 - x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1\)
\(13\)	\( 1 + 2 T + 13 T^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)	\( ( 1 - 5 T^{2} )^{2}\)
\(7\)	\( ( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )\)
\(11\)	\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}\)
\(17\)	\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}\)
\(19\)	\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)	\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)	\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

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.