Properties

Label 17689.a
Conductor $17689$
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

Learn more about

Genus 2 curves in isogeny class 17689.a

Label Equation
17689.a.17689.1 \(y^2 + (x^2 + x)y = x^5 - 4x^4 + 2x^3 + x^2 - x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(7\)\( 1 + 4 T + 7 T^{2}\)
\(19\)\( 1 - 8 T + 19 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 - T^{2} + 4 T^{4}\)
\(3\)\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - 5 T^{2} )^{2}\)
\(11\)\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )\)
\(17\)\( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}\)
\(23\)\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}\)
$\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 \(K = \Q (a) \simeq \) 6.6.41615795893.1 with defining polynomial \(x^{6} - x^{5} - 55 x^{4} + 160 x^{3} - 20 x^{2} - 176 x + 64\)

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.