Genus 2 isogeny class 13689.a

• Label: 13689.a
• Conductor: $13689$
• 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 13689.a

Label	Equation
13689.a.13689.1	\(y^2 + (x^3 + x + 1)y = x^5 - x^4 - 4x^3 - 2x^2\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(3\)	\( 1 + 3 T + 3 T^{2}\)
\(13\)	\( 1 + 5 T + 13 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(5\)	\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)	\( ( 1 - 5 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)	\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}\)
\(17\)	\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}\)
\(19\)	\( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4}\)
\(23\)	\( ( 1 + 3 T + 23 T^{2} )^{2}\)
\(29\)	\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.6.2436053373.2 with defining polynomial:
  \(x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117\)

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
  \(g_4 = -\frac{12015765}{3956} b^{5} + \frac{233936235}{15824} b^{4} + \frac{735217587}{15824} b^{3} - \frac{2326501125}{15824} b^{2} - \frac{240909201}{688} b - \frac{288784197}{3956}\)
  \(g_6 = \frac{36685451205}{3956} b^{5} - \frac{2858709285549}{63296} b^{4} - \frac{8968980313215}{63296} b^{3} + \frac{28420667571231}{63296} b^{2} + \frac{2939506491519}{2752} b + \frac{14100757396245}{63296}\)
   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 \(K = \Q (a) \simeq \) 6.6.2436053373.2 with defining polynomial \(x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117\)

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.