# Properties

 Label 100315.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 100315.a

Label Equation
100315.a.501575.1 $$y^2 + (x^3 + x + 1)y = 2x^5 + 3x^4 + x^3 + 2x^2 - 5x + 1$$

## L-function data

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

Prime L-Factor
$$5$$$$( 1 - T )( 1 + T + 5 T^{2} )$$
$$20063$$$$( 1 - T )( 1 - 88 T + 20063 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + T + 3 T^{2} + 3 T^{3} + 9 T^{4}$$
$$7$$$$1 + 2 T + 9 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 + T + 7 T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 + 8 T + 38 T^{2} + 136 T^{3} + 289 T^{4}$$
$$19$$$$1 + 11 T + 67 T^{2} + 209 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 5 T + 23 T^{2} )( 1 + 3 T + 23 T^{2} )$$
$$29$$$$1 + 2 T + 48 T^{2} + 58 T^{3} + 841 T^{4}$$
$\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.