Genus 2 Isogeny Class 20172.b

• Label: 20172.b
• Sato-Tate group: $G_{3,3}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 20172.b

Label	Equation
20172.b.968256.1	\(y^2 + (x^2 + x)y = x^6 - x^5 - 2x^4 + 4x^3 - 3x + 1\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( ( 1 + T )^{2}\)
\(3\)	\( ( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(41\)	\( ( 1 + T )^{2}\)

Good L-factors:

Prime	L-Factor
\(5\)	\( ( 1 + 2 T + 5 T^{2} )^{2}\)
\(7\)	\( ( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\( ( 1 + 2 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(13\)	\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)	\( ( 1 + 2 T + 17 T^{2} )^{2}\)
\(19\)	\( ( 1 - 6 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)	\( ( 1 - 4 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)	\( ( 1 + 29 T^{2} )( 1 + 8 T + 29 T^{2} )\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $G_{3,3}$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

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

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.