Properties

 Label 3469.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 3469.a

Label Equation
3469.a.3469.1 $$y^2 + (x^3 + x + 1)y = x^5 - x^4 - 2x^3$$

L-function data

Analytic rank:$$2$$

Prime L-Factor
$$3469$$$$( 1 + T )( 1 - 25 T + 3469 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$5$$$$( 1 + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$1 + 5 T + 15 T^{2} + 35 T^{3} + 49 T^{4}$$
$$11$$$$1 + 3 T + 19 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$1 - 2 T - 7 T^{2} - 26 T^{3} + 169 T^{4}$$
$$17$$$$1 + 5 T + 31 T^{2} + 85 T^{3} + 289 T^{4}$$
$$19$$$$1 + 4 T + 18 T^{2} + 76 T^{3} + 361 T^{4}$$
$$23$$$$1 + 5 T^{2} + 529 T^{4}$$
$$29$$$$1 + 20 T^{2} + 841 T^{4}$$
$\cdots$$\cdots$

Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

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.