# Properties

 Label 6463.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 6463.a

Label Equation
6463.a.6463.1 $$y^2 + (x^2 + x + 1)y = x^6 - x^4$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$23$$$$( 1 + T )( 1 + 3 T + 23 T^{2} )$$
$$281$$$$( 1 + T )( 1 - 10 T + 281 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4}$$
$$7$$$$1 + 2 T + 13 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 + 3 T + 19 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$1 - 2 T + 9 T^{2} - 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - 13 T^{2} + 289 T^{4}$$
$$19$$$$1 + 7 T + 31 T^{2} + 133 T^{3} + 361 T^{4}$$
$$29$$$$1 + 7 T + 25 T^{2} + 203 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$
Not of $$\GL_2$$-type over $$\Q$$
All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.