Genus 2 isogeny class 1042.a

• Label: 1042.a
• Conductor: $1042$
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 1042.a

Label	Equation
1042.a.1042.1	\(y^2 + (x^3 + x)y = -x^4 - x^3 - x^2 + 2x + 2\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( ( 1 + T )( 1 + 2 T^{2} )\)
\(521\)	\( ( 1 + T )( 1 + 27 T + 521 T^{2} )\)

Good L-factors:

Prime	L-Factor
\(3\)	\( 1 - T + 3 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)	\( ( 1 - 4 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)	\( ( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\( 1 - 2 T - 8 T^{2} - 22 T^{3} + 121 T^{4}\)
\(13\)	\( 1 - 2 T + 20 T^{2} - 26 T^{3} + 169 T^{4}\)
\(17\)	\( ( 1 - 6 T + 17 T^{2} )( 1 + 3 T + 17 T^{2} )\)
\(19\)	\( 1 + T + 23 T^{2} + 19 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 6 T + 28 T^{2} + 138 T^{3} + 529 T^{4}\)
\(29\)	\( 1 + 6 T + 22 T^{2} + 174 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.