Properties

 Label 59107.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

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Genus 2 curves in isogeny class 59107.a

Label Equation
59107.a.59107.1 $$y^2 + (x^3 + x^2 + 1)y = 2x^4 - 3x^2 - x$$

L-function data

Analytic rank:$$3$$  (upper bound)
Mordell-Weil rank:$$3$$

Bad L-factors:
Prime L-Factor
$$59107$$$$( 1 - T )( 1 - 200 T + 59107 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$5$$$$( 1 + T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$$11$$$$1 + 7 T + 31 T^{2} + 77 T^{3} + 121 T^{4}$$
$$13$$$$1 + 7 T + 29 T^{2} + 91 T^{3} + 169 T^{4}$$
$$17$$$$1 + 2 T + 12 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$$23$$$$1 + 5 T + 16 T^{2} + 115 T^{3} + 529 T^{4}$$
$$29$$$$1 - T - 8 T^{2} - 29 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

See L-function page for more information

Sato-Tate group

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

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.