Properties

Label 12544.i
Conductor $12544$
Sato-Tate group $E_3$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 12544.i

Label Equation
12544.i.614656.1 \(y^2 = x^5 + 4x^4 - 13x^3 + 9x^2 - x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(7\)\( 1 - 4 T + 7 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(11\)\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )^{2}\)
\(17\)\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)\( ( 1 + 6 T + 29 T^{2} )^{2}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_3$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial:
  \(x^{3} - x^{2} - 2 x + 1\)

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
  \(g_4 = -15349824 b^{2} + 8518608 b + 34491744\)
  \(g_6 = 17221522752 b^{2} - 9557224992 b - 38696416416\)
   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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial \(x^{3} - x^{2} - 2 x + 1\)

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.