Properties

Label 52441.a
Conductor $52441$
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 52441.a

Label Equation
52441.a.52441.1 \(y^2 + (x^2 + x)y = x^5 + 5x^4 + 5x^3 - x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(229\)\( 1 - 22 T + 229 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 - T^{2} + 4 T^{4}\)
\(3\)\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( ( 1 + 11 T^{2} )^{2}\)
\(13\)\( ( 1 - 13 T^{2} )^{2}\)
\(17\)\( ( 1 - 6 T + 17 T^{2} )^{2}\)
\(19\)\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.629763392149.1 with defining polynomial \(x^{6} - x^{5} - 95 x^{4} + 530 x^{3} - 925 x^{2} + 367 x + 187\)

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.