Properties

Label 3969.b
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 3969.b

Label Equation
3969.b.35721.1 \(y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - 3x^2\)

L-function data

Analytic rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( 1 + 3 T + 3 T^{2}\)
\(7\)\( 1 + 5 T + 7 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(5\)\( ( 1 + 3 T + 5 T^{2} )^{2}\)
\(11\)\( 1 - 19 T^{2} + 121 T^{4}\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )\)
\(17\)\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( 1 - 19 T^{2} + 529 T^{4}\)
\(29\)\( 1 + 9 T + 56 T^{2} + 261 T^{3} + 841 T^{4}\)
$\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 \(\Q\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.330812181.1 with defining polynomial \(x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35\)

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.