Genus 2 isogeny class 51984.a

• Label: 51984.a
• Conductor: $51984$
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 51984.a

Label	Equation
51984.a.467856.1	\(y^2 + x^3y = x^4 + 2x^2 - 1\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T + 2 T^{2}\)
\(3\)	\( ( 1 - T )( 1 + T )\)
\(19\)	\( ( 1 + T )^{2}\)

Good L-factors:

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

See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 912.f
  Elliptic curve isogeny class 57.a

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.