Genus 2 isogeny class 727609.a

• Label: 727609.a
• Conductor: $727609$
• 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

Genus 2 curves in isogeny class 727609.a

Label	Equation
727609.a.727609.1	\(y^2 + (x^3 + x + 1)y = -3x^5 + 11x^4 - 11x^3 + x\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(853\)	\( 1 - 35 T + 853 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(3\)	\( ( 1 + 2 T + 3 T^{2} )^{2}\)
\(5\)	\( 1 - 7 T^{2} + 25 T^{4}\)
\(7\)	\( ( 1 - 7 T^{2} )^{2}\)
\(11\)	\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}\)
\(13\)	\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )\)
\(17\)	\( 1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4}\)
\(19\)	\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}\)
\(23\)	\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}\)
\(29\)	\( 1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

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 \(\Q (b)\) with defining polynomial:
  \(x^{6} - x^{5} - 355 x^{4} + 2164 x^{3} + 12951 x^{2} - 127278 x + 250479\)

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
  \(g_4 = -\frac{91753778777}{1310256} b^{5} - \frac{427172560315}{1310256} b^{4} + \frac{15100223464969}{655128} b^{3} - \frac{26896534681157}{1310256} b^{2} - \frac{223789793030201}{218376} b + \frac{224889490077251}{72792}\)
  \(g_6 = \frac{3723001140950777}{5241024} b^{5} + \frac{16647437391232375}{5241024} b^{4} - \frac{309666823897929449}{1310256} b^{3} + \frac{1131323133896869361}{5241024} b^{2} + \frac{9177326539137560825}{873504} b - \frac{18418535473094813119}{582336}\)
   Conductor norm: 1

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 \(K = \Q (a)\) with defining polynomial \(x^{6} - x^{5} - 355 x^{4} + 2164 x^{3} + 12951 x^{2} - 127278 x + 250479\)

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.