Properties

 Label 971.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 971.a

Label Equation
971.a.971.1 $$y^2 + y = x^5 - 2x^3 + x$$

L-function data

Analytic rank:$$1$$

Prime L-Factor
$$971$$$$( 1 - T )( 1 + 45 T + 971 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}$$
$$5$$$$1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$1 - T + T^{2} - 7 T^{3} + 49 T^{4}$$
$$11$$$$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$$
$$13$$$$1 - T - 8 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$1 - T + 3 T^{2} - 19 T^{3} + 361 T^{4}$$
$$23$$$$1 - T + 35 T^{2} - 23 T^{3} + 529 T^{4}$$
$$29$$$$1 + T - 18 T^{2} + 29 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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