Properties

Label 448.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 448.a

Label Equation
448.a.448.2 \(y^2 + (x^3 + x)y = -2x^4 + 7\)
448.a.448.1 \(y^2 + (x^3 + x)y = x^4 - 7\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T\)
\(7\)\( ( 1 - T )( 1 + 7 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(5\)\( ( 1 + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)
\(11\)\( ( 1 + 11 T^{2} )^{2}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )( 1 - 2 T + 17 T^{2} )\)
\(19\)\( ( 1 - 2 T + 19 T^{2} )( 1 + 19 T^{2} )\)
\(23\)\( ( 1 + 23 T^{2} )^{2}\)
\(29\)\( ( 1 + 6 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)
$\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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

Endomorphism algebra over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-1}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)

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