Properties

 Label 4312.a Conductor $4312$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Related objects

Genus 2 curves in isogeny class 4312.a

Label Equation
4312.a.189728.1 $$y^2 + (x + 1)y = -2x^5 + 4x^4 - 2x^3 - 2x^2$$
4312.a.551936.1 $$y^2 + (x^3 + x)y = x^4 - 2x^3 + 2x^2 - 2x$$

L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$2$$$$1 - T$$
$$7$$$$1 + T^{2}$$
$$11$$$$( 1 + T )( 1 - 2 T + 11 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$13$$$$1 + 169 T^{4}$$
$$17$$$$1 + 6 T + 22 T^{2} + 102 T^{3} + 289 T^{4}$$
$$19$$$$1 + 2 T + 18 T^{2} + 38 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$1 - 2 T^{2} + 841 T^{4}$$
$\cdots$$\cdots$

Sato-Tate group

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

Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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.