Genus 2 isogeny class 8649.b

• Label: 8649.b
• Conductor: $8649$
• 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 8649.b

Label	Equation
8649.b.700569.1	\(y^2 + (x^2 + x)y = 9x^5 + 2x^4 - 21x^3 - 22x^2 - 8x - 1\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(3\)	\( 1 - 3 T + 3 T^{2}\)
\(31\)	\( 1 - 4 T + 31 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 - T^{2} + 4 T^{4}\)
\(5\)	\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)	\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}\)
\(13\)	\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)	\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}\)
\(19\)	\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}\)
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)
\(29\)	\( ( 1 - 6 T + 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.6.772987077.1 with defining polynomial:
  \(x^{6} - x^{5} - 28 x^{4} + 51 x^{3} + 75 x^{2} - 98 x - 92\)

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
  \(g_4 = -\frac{16934823333}{3104} b^{5} + \frac{15604267599}{1552} b^{4} + \frac{223933318713}{1552} b^{3} - \frac{1241168535621}{3104} b^{2} - \frac{111941475201}{1552} b + \frac{462072237363}{776}\)
  \(g_6 = -\frac{579872195059509}{6208} b^{5} + \frac{1068664047901587}{6208} b^{4} + \frac{15335612391671001}{6208} b^{3} - \frac{2656270916385201}{388} b^{2} - \frac{958200176670183}{776} b + \frac{7911131565422907}{776}\)
   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 \(K = \Q (a) \simeq \) 6.6.772987077.1 with defining polynomial \(x^{6} - x^{5} - 28 x^{4} + 51 x^{3} + 75 x^{2} - 98 x - 92\)

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.