Properties

Label 971.a
Conductor 971971
Sato-Tate group USp(4)\mathrm{USp}(4)
End(JQ)R\End(J_{\overline{\Q}}) \otimes \R R\R
End(JQ)Q\End(J_{\overline{\Q}}) \otimes \Q Q\Q
End(J)Q\End(J) \otimes \Q Q\Q
Q\overline{\Q}-simple yes
GL2\mathrm{GL}_2-type no

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Genus 2 curves in isogeny class 971.a

Label Equation
971.a.971.1 y2+y=x52x3+xy^2 + y = x^5 - 2x^3 + x

L-function data

Analytic rank:11
Mordell-Weil rank:11
 
Bad L-factors:
Prime L-Factor
971971(1T)(1+45T+971T2) ( 1 - T )( 1 + 45 T + 971 T^{2} )
 
Good L-factors:
Prime L-Factor
22(1+2T2)(1+2T+2T2) ( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )
331+3T+5T2+9T3+9T4 1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}
551+3T+11T2+15T3+25T4 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4}
771T+T27T3+49T4 1 - T + T^{2} - 7 T^{3} + 49 T^{4}
11111+5T+14T2+55T3+121T4 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}
13131T8T213T3+169T4 1 - T - 8 T^{2} - 13 T^{3} + 169 T^{4}
17171+2T+2T2+34T3+289T4 1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}
19191T+3T219T3+361T4 1 - T + 3 T^{2} - 19 T^{3} + 361 T^{4}
23231T+35T223T3+529T4 1 - T + 35 T^{2} - 23 T^{3} + 529 T^{4}
29291+T18T2+29T3+841T4 1 + T - 18 T^{2} + 29 T^{3} + 841 T^{4}
\cdots\cdots
 
See L-function page for more information

Sato-Tate group

ST=\mathrm{ST} = USp(4)\mathrm{USp}(4)

Decomposition of the Jacobian

Simple over Q\overline{\Q}

Endomorphisms of the Jacobian

Not of GL2\GL_2-type over Q\Q

Endomorphism algebra over Q\Q:

End(J)Q\End (J_{}) \otimes \Q \simeqQ\Q
End(J)R\End (J_{}) \otimes \R\simeq R\R

All Q\overline{\Q}-endomorphisms of the Jacobian are defined over Q\Q.

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.