# Properties

 Label 529.a Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 529.a

Label Equation
529.a.529.1 $$y^2 + (x^3 + x + 1)y = -x^5$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$23$$$$( 1 - T )^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + T^{2} + 9 T^{4}$$
$$5$$$$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4}$$
$$11$$$$1 + 6 T + 26 T^{2} + 66 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 3 T + 13 T^{2} )^{2}$$
$$17$$$$1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4}$$
$$19$$$$( 1 + 2 T + 19 T^{2} )^{2}$$
$$29$$$$( 1 + 3 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

$$\mathrm{ST} =$$ $G_{3,3}$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)$$
Of $$\GL_2$$-type over $$\Q$$
All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.