Properties

Label 10048.b
Conductor 1004810048
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 10048.b

Label Equation
10048.b.160768.1 y2+(x3+x2)y=x4+3x3+5x2+4x+2y^2 + (x^3 + x^2)y = x^4 + 3x^3 + 5x^2 + 4x + 2

L-function data

Analytic rank:11
Mordell-Weil rank:11
 
Bad L-factors:
Prime L-Factor
221T 1 - T
157157(1T)(1+7T+157T2) ( 1 - T )( 1 + 7 T + 157 T^{2} )
 
Good L-factors:
Prime L-Factor
331+2T+5T2+6T3+9T4 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4}
551+3T+4T2+15T3+25T4 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}
77(1+7T2)(1+4T+7T2) ( 1 + 7 T^{2} )( 1 + 4 T + 7 T^{2} )
111114T+11T244T3+121T4 1 - 4 T + 11 T^{2} - 44 T^{3} + 121 T^{4}
13131+8T+37T2+104T3+169T4 1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4}
17171+2T+15T2+34T3+289T4 1 + 2 T + 15 T^{2} + 34 T^{3} + 289 T^{4}
191914T+19T276T3+361T4 1 - 4 T + 19 T^{2} - 76 T^{3} + 361 T^{4}
23231+2T+10T2+46T3+529T4 1 + 2 T + 10 T^{2} + 46 T^{3} + 529 T^{4}
292912T2+841T4 1 - 2 T^{2} + 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.