Properties

Label 21316.a
Conductor $21316$
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 21316.a

Label Equation
21316.a.42632.1 \(y^2 + (x^3 + x + 1)y = 3x^3 + 4x^2 + x\)

L-function data

Analytic rank:\(2\)  (upper bound)
Mordell-Weil rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T + T^{2}\)
\(73\)\( 1 + 17 T + 73 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 + T + 3 T^{2} )^{2}\)
\(5\)\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 6 T + 25 T^{2} + 78 T^{3} + 169 T^{4}\)
\(17\)\( ( 1 - 17 T^{2} )^{2}\)
\(19\)\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4}\)
$\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 \(K = \Q (a) \simeq \) 6.6.2073071593.1 with defining polynomial \(x^{6} - x^{5} - 30 x^{4} + 31 x^{3} + 206 x^{2} - 150 x - 81\)

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.