# Properties

 Label 10110.a Conductor $10110$ 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 10110.a

Label Equation
10110.a.50550.2 $$y^2 + (x^2 + x)y = 39x^6 - 45x^5 - 62x^4 + 22x^3 + 69x^2 + 50x - 75$$
10110.a.50550.1 $$y^2 + (x^3 + 1)y = 2x^5 + 3x^4 + 2x^3 - x - 1$$

## L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Prime L-Factor
$$2$$$$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$$$( 1 + T )( 1 - T + 3 T^{2} )$$
$$5$$$$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$337$$$$( 1 - T )( 1 + 14 T + 337 T^{2} )$$

Good L-factors:
Prime L-Factor
$$7$$$$1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 - 4 T + 14 T^{2} - 44 T^{3} + 121 T^{4}$$
$$13$$$$( 1 + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$1 + 2 T^{2} + 289 T^{4}$$
$$19$$$$1 + 10 T^{2} + 361 T^{4}$$
$$23$$$$1 + T - 18 T^{2} + 23 T^{3} + 529 T^{4}$$
$$29$$$$1 + 9 T + 40 T^{2} + 261 T^{3} + 841 T^{4}$$
$\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.