Properties

Label 3721.a
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 3721.a

Label Equation
3721.a.3721.1 \(y^2 + (x^3 + x + 1)y = -x^4 + x^3 + 3x^2 + x\)

L-function data

Analytic rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(61\)\( 1 + T + 61 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(3\)\( ( 1 + 2 T + 3 T^{2} )^{2}\)
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)\( 1 - 10 T^{2} + 121 T^{4}\)
\(13\)\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )\)
\(17\)\( 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4}\)
\(19\)\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 23 T^{2} )^{2}\)
\(29\)\( 1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4}\)
$\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 \(\Q\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.844596301.1 with defining polynomial \(x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27\)

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.