Properties

Label 72900.a
Conductor 7290072900
Sato-Tate group J(E6)J(E_6)
End(JQ)R\End(J_{\overline{\Q}}) \otimes \R M2(R)\mathrm{M}_2(\R)
End(JQ)Q\End(J_{\overline{\Q}}) \otimes \Q M2(Q)\mathrm{M}_2(\Q)
End(J)Q\End(J) \otimes \Q Q\Q
Q\overline{\Q}-simple no
GL2\mathrm{GL}_2-type no

Related objects

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Genus 2 curves in isogeny class 72900.a

Label Equation
72900.a.291600.1 y2+y=x62x3+1y^2 + y = x^6 - 2x^3 + 1

L-function data

Analytic rank:11
Mordell-Weil rank:11
 
Bad L-factors:
Prime L-Factor
221+2T2 1 + 2 T^{2}
331 1
551+5T2 1 + 5 T^{2}
 
Good L-factors:
Prime L-Factor
771+7T2+49T4 1 + 7 T^{2} + 49 T^{4}
1111(13T+11T2)(1+3T+11T2) ( 1 - 3 T + 11 T^{2} )( 1 + 3 T + 11 T^{2} )
1313(12T+13T2)(1+5T+13T2) ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )
1717(1+17T2)2 ( 1 + 17 T^{2} )^{2}
19191+2T15T2+38T3+361T4 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}
23231+19T2+529T4 1 + 19 T^{2} + 529 T^{4}
2929(16T+29T2)(1+6T+29T2) ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )
\cdots\cdots
 
See L-function page for more information

Sato-Tate group

ST=\mathrm{ST} = J(E6)J(E_6), ST0=SU(2)\quad \mathrm{ST}^0 = \mathrm{SU}(2)

Decomposition of the Jacobian

Splits over the number field Q(b)\Q (b) \simeq 6.2.450000.1 with defining polynomial:
  x6x55x3x+1x^{6} - x^{5} - 5 x^{3} - x + 1

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  y2=x3g4/48xg6/864y^2 = x^3 - g_4 / 48 x - g_6 / 864 with
  g4=192b5690b4+690b3750b2+2280b672g_4 = 192 b^{5} - 690 b^{4} + 690 b^{3} - 750 b^{2} + 2280 b - 672
  g6=20160b5+10080b4+20160b3+80640b2+40320b39060g_6 = -20160 b^{5} + 10080 b^{4} + 20160 b^{3} + 80640 b^{2} + 40320 b - 39060
   Conductor norm: 4782969
  y2=x3g4/48xg6/864y^2 = x^3 - g_4 / 48 x - g_6 / 864 with
  g4=192b5210b4+210b3+1650b2+1320b+672g_4 = -192 b^{5} - 210 b^{4} + 210 b^{3} + 1650 b^{2} + 1320 b + 672
  g6=20160b510080b420160b380640b240320b28980g_6 = 20160 b^{5} - 10080 b^{4} - 20160 b^{3} - 80640 b^{2} - 40320 b - 28980
   Conductor norm: 4782969

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

Smallest field over which all endomorphisms are defined:
Galois number field K=Q(a)K = \Q (a) with defining polynomial x12x11+x10+10x95x8x7+26x6x55x4+10x3+x2x+1x^{12} - x^{11} + x^{10} + 10 x^{9} - 5 x^{8} - x^{7} + 26 x^{6} - x^{5} - 5 x^{4} + 10 x^{3} + x^{2} - x + 1

Endomorphism algebra over Q\overline{\Q}:

End(JQ)Q\End (J_{\overline{\Q}}) \otimes \Q \simeqM2(\mathrm{M}_2(Q\Q))
End(JQ)R\End (J_{\overline{\Q}}) \otimes \R\simeq M2(R)\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.