Properties

Label 6400.f
Sato-Tate group $E_4$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 6400.f

Label Equation
6400.f.64000.1 \(y^2 + x^3y = -2x^4 - 3x^3 + x^2 + 6x + 4\)

L-function data

Analytic rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 2 T^{2}\)
\(5\)\( 1 + 4 T + 5 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 + 2 T + 3 T^{2} )^{2}\)
\(7\)\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4}\)
\(13\)\( 1 - 22 T^{2} + 169 T^{4}\)
\(17\)\( 1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4}\)
\(23\)\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 + 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_4$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Endomorphisms of the Jacobian

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 4.4.256000.1 with defining polynomial \(x^{4} - 20 x^{2} + 10\)

Endomorphism algebra over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

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