# Properties

 Label 7529.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 7529.a

Label Equation
7529.a.7529.1 $$y^2 + (x^3 + x^2 + x)y = x^2 + 2x + 1$$

## L-function data

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

Prime L-Factor
$$7529$$$$( 1 + T )( 1 + 15 T + 7529 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$5$$$$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$1 + 4 T + 23 T^{2} + 44 T^{3} + 121 T^{4}$$
$$13$$$$1 + 3 T + 4 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 + 4 T + 25 T^{2} + 68 T^{3} + 289 T^{4}$$
$$19$$$$( 1 + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 2 T - 5 T^{2} - 46 T^{3} + 529 T^{4}$$
$$29$$$$1 - 7 T + 46 T^{2} - 203 T^{3} + 841 T^{4}$$
$\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.