Genus 2 Isogeny Class 19881.b

• Label: 19881.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 19881.b

Label	Equation
19881.b.536787.1	\(y^2 + y = x^6 - 3x^5 + 2x^4 + x^3 - 2x^2 + x\)

L-function data

Analytic rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(3\)	\( ( 1 - T )( 1 + T )\)
\(47\)	\( ( 1 - T )( 1 + T )\)

Good L-factors:

Prime	L-Factor
\(2\)	\( ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )\)
\(5\)	\( ( 1 + T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)	\( ( 1 + 3 T + 7 T^{2} )^{2}\)
\(11\)	\( ( 1 + 3 T + 11 T^{2} )( 1 + 5 T + 11 T^{2} )\)
\(13\)	\( ( 1 - 2 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)	\( ( 1 - 8 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )\)
\(19\)	\( ( 1 + 6 T + 19 T^{2} )^{2}\)
\(23\)	\( ( 1 - 9 T + 23 T^{2} )( 1 - 3 T + 23 T^{2} )\)
\(29\)	\( ( 1 - T + 29 T^{2} )( 1 + T + 29 T^{2} )\)
$\cdots$	$\cdots$

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 \(\Q\)

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.