Properties

Label 331776.g
Conductor $331776$
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 331776.g

Label Equation
331776.g.995328.1 \(y^2 = x^5 - 3x^3 + 3x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(3\)\( 1\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4}\)
\(7\)\( 1 + 6 T^{2} + 49 T^{4}\)
\(11\)\( ( 1 - 2 T + 11 T^{2} )( 1 + 2 T + 11 T^{2} )\)
\(13\)\( ( 1 + 4 T + 13 T^{2} )^{2}\)
\(17\)\( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 30 T^{2} + 361 T^{4}\)
\(23\)\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )\)
\(29\)\( 1 - 4 T + 8 T^{2} - 116 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_4)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.2.6912.1 with defining polynomial:
  \(x^{4} - 3\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 144\)
  \(g_6 = -1792 b^{3} - 3456 b\)
   Conductor norm: 2304
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 144\)
  \(g_6 = 1792 b^{3} + 3456 b\)
   Conductor norm: 2304

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.47775744.1 with defining polynomial \(x^{8} + 3 x^{4} + 9\)

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.