Properties

Label 864.a
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM} \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 864.a

Label Equation
864.a.1728.1 \(y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^2\)
864.a.221184.1 \(y^2 + x^3y = x^5 - 4x^4 - 6x^3 + 33x^2 - 36x + 12\)
864.a.442368.1 \(y^2 = x^6 - 4x^4 + 6x^2 - 3\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(3\)\( 1 + T\)
 
Good L-factors:
Prime L-Factor
\(5\)\( ( 1 + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)
\(7\)\( ( 1 + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( ( 1 - 4 T + 11 T^{2} )( 1 + 11 T^{2} )\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )( 1 + 17 T^{2} )\)
\(19\)\( ( 1 - 8 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( ( 1 + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $N(G_{1,3})$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{SU}(2)\)

Endomorphisms of the Jacobian

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Endomorphism algebra over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)

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