# Properties

 Label 961.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 961.a

Label Equation
961.a.961.1 $$y^2 + (x^3 + x + 1)y = -x^6 - x^5 - 7x^4 + 74x^3 - 145x^2 + 99x - 33$$
961.a.961.2 $$y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 8x^4 + 12x^3 - 18x^2 + 12x - 7$$
961.a.961.3 $$y^2 + (x^3 + x + 1)y = x^5 + x^4 + x^3 - x - 1$$
961.a.923521.1 $$y^2 + (x^3 + x^2 + 1)y = -5x^4 + 4x^3 + 3x^2 - 2x - 3$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$31$$$$( 1 - T )^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - T + 5 T^{2} )^{2}$$
$$7$$$$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 2 T + 11 T^{2} )^{2}$$
$$13$$$$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4}$$
$$19$$$$1 + 33 T^{2} + 361 T^{4}$$
$$23$$$$1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$1 - 10 T + 78 T^{2} - 290 T^{3} + 841 T^{4}$$
$\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 $$\Q$$.