# Properties

 Label 100036.a Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 100036.a

Label Equation
100036.a.200072.1 $$y^2 + (x^3 + x^2 + 1)y = x^5 - 8x^3 - x^2 + 12x - 6$$

## L-function data

Analytic rank:$$2$$  (upper bound)
Mordell-Weil rank:$$2$$

Prime L-Factor
$$2$$$$1 + T + T^{2}$$
$$89$$$$( 1 + T )( 1 - 14 T + 89 T^{2} )$$
$$281$$$$( 1 + T )( 1 - 20 T + 281 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 2 T + 3 T^{2} )^{2}$$
$$5$$$$1 + 5 T^{2} + 25 T^{4}$$
$$7$$$$1 + T + 6 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$1 + 14 T^{2} + 121 T^{4}$$
$$13$$$$( 1 + T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$$17$$$$( 1 + 2 T + 17 T^{2} )^{2}$$
$$19$$$$1 - 2 T - 11 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$1 - 4 T - 6 T^{2} - 92 T^{3} + 529 T^{4}$$
$$29$$$$1 + 4 T + 32 T^{2} + 116 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## 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.