Genus 2 isogeny class 3264.a

• Label: 3264.a
• Conductor: $3264$
• Sato-Tate group: $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 3264.a

Label	Equation
3264.a.3264.1	\(y^2 + x^3y = x^5 - 3x^3 + 4x - 2\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T + 2 T^{2}\)
\(3\)	\( ( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(17\)	\( ( 1 + T )( 1 + 6 T + 17 T^{2} )\)

Good L-factors:

Prime	L-Factor
\(5\)	\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)	\( 1 + 2 T^{2} + 49 T^{4}\)
\(11\)	\( ( 1 + 2 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(13\)	\( 1 - 6 T^{2} + 169 T^{4}\)
\(19\)	\( ( 1 + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)	\( 1 + 6 T^{2} + 529 T^{4}\)
\(29\)	\( 1 - 46 T^{2} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-2}) \) with defining polynomial:
  \(x^{2} + 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.0.8.1-51.4-a
  Elliptic curve isogeny class 2.0.8.1-51.1-a

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-2}) \) with defining polynomial \(x^{2} + 2\)

Endomorphism algebra over \(\overline{\Q}\):

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

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.