Properties

Label 12544.g
Conductor $12544$
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 12544.g

Label Equation
12544.g.175616.1 \(y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(7\)\( 1 + 7 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 8 T^{2} + 25 T^{4}\)
\(11\)\( 1 - 6 T^{2} + 121 T^{4}\)
\(13\)\( 1 + 8 T^{2} + 169 T^{4}\)
\(17\)\( 1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 4 T + 8 T^{2} + 76 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )\)
\(29\)\( ( 1 - 8 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )\)
$\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 \(\Q (b) \simeq \) 4.0.2744.1 with defining polynomial:
  \(x^{4} - x^{3} + 3 x^{2} + 3 x + 2\)

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
  \(g_4 = -56 b^{3} + 112 b^{2} + 280 b + 112\)
  \(g_6 = -812 b^{3} + 8596 b^{2} + 7700 b + 5124\)
   Conductor norm: 1024
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -280 b^{3} + 560 b^{2} - 1288 b + 112\)
  \(g_6 = -16338 b^{3} + 25704 b^{2} - 59150 b - 22764\)
   Conductor norm: 1024

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 \(K = \Q (a) \simeq \) 8.0.481890304.3 with defining polynomial \(x^{8} - 2 x^{7} - 3 x^{6} + 8 x^{5} + x^{4} - 2 x^{3} + 23 x^{2} - 12 x + 2\)

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.