Properties

 Label 800320.a Conductor $800320$ 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

Related objects

Genus 2 curves in isogeny class 800320.a

Label Equation
800320.a.800320.1 $$y^2 + (x^3 + x^2 + x + 1)y = x^6 + 2x^4 - 3x^3 - 3x + 1$$

L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$2$$$$1 - 2 T + 2 T^{2}$$
$$5$$$$( 1 + T )( 1 - 4 T + 5 T^{2} )$$
$$41$$$$( 1 + T )( 1 + 12 T + 41 T^{2} )$$
$$61$$$$( 1 - T )( 1 - 7 T + 61 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4}$$
$$7$$$$1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - T + 11 T^{2} )( 1 + 5 T + 11 T^{2} )$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$1 - 3 T + 28 T^{2} - 51 T^{3} + 289 T^{4}$$
$$19$$$$1 + 26 T^{2} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$$
$$29$$$$( 1 - 2 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

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.