Properties

Label 92416.b
Conductor $92416$
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 92416.b

Label Equation
92416.b.92416.1 \(y^2 = x^5 + x^4 - 7x^3 + 6x^2 - x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(19\)\( 1 - 8 T + 19 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)\( ( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)\( 1 - 10 T^{2} + 121 T^{4}\)
\(13\)\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}\)
\(23\)\( 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 15 T + 104 T^{2} - 435 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.158470336.1 with defining polynomial:
  \(x^{6} - 19 x^{4} + 38 x^{2} - 19\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{840320}{7} b^{5} - \frac{988912}{7} b^{4} + \frac{14804800}{7} b^{3} + \frac{17429232}{7} b^{2} - \frac{11474784}{7} b - \frac{13559312}{7}\)
  \(g_6 = -\frac{592119040}{7} b^{5} - \frac{690754272}{7} b^{4} + \frac{10444093888}{7} b^{3} + \frac{12182781152}{7} b^{2} - \frac{8281413568}{7} b - \frac{9648237696}{7}\)
   Conductor norm: 4096

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.158470336.1 with defining polynomial \(x^{6} - 19 x^{4} + 38 x^{2} - 19\)

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.