Genus 2 isogeny class 27200.d

• Label: 27200.d
• Conductor: $27200$
• 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 27200.d

Label	Equation
27200.d.680000.1	\(y^2 + x^3y = x^4 - x^3 + 2x^2 + 2\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T + 2 T^{2}\)
\(5\)	\( ( 1 - T )( 1 + T )\)
\(17\)	\( ( 1 + T )( 1 + 2 T + 17 T^{2} )\)

Good L-factors:

Prime	L-Factor
\(3\)	\( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(7\)	\( ( 1 + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)	\( 1 + 10 T^{2} + 121 T^{4}\)
\(13\)	\( 1 - 14 T^{2} + 169 T^{4}\)
\(19\)	\( 1 - 18 T^{2} + 361 T^{4}\)
\(23\)	\( ( 1 - 8 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)
\(29\)	\( 1 - 26 T^{2} + 841 T^{4}\)
$\cdots$	$\cdots$

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 2.2.8.1-425.2-d
  Elliptic curve isogeny class 2.2.8.1-425.1-d

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.