Genus 2 isogeny class 613089.b

• Label: 613089.b
• Conductor: $613089$
• Sato-Tate group: $J(E_6)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 613089.b

Label	Equation
613089.b.613089.1	\(y^2 + (x^3 + 1)y = 2x^3 + 7\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(3\)	\( 1\)
\(29\)	\( 1 + 29 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 + T^{2} + 4 T^{4}\)
\(5\)	\( ( 1 + 5 T^{2} )^{2}\)
\(7\)	\( ( 1 - 2 T + 7 T^{2} )^{2}\)
\(11\)	\( 1 + 19 T^{2} + 121 T^{4}\)
\(13\)	\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)	\( 1 + 22 T^{2} + 289 T^{4}\)
\(19\)	\( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4}\)
\(23\)	\( ( 1 - 3 T + 23 T^{2} )( 1 + 3 T + 23 T^{2} )\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_6)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b)\) with defining polynomial:
  \(x^{6} - 29\)

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
  \(g_4 = 180 b^{5} + 1305 b^{2}\)
  \(g_6 = -54810 b^{3} - 288927\)
   Conductor norm: 729
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -180 b^{5} + 1305 b^{2}\)
  \(g_6 = 54810 b^{3} - 288927\)
   Conductor norm: 729

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 \(K = \Q (a)\) with defining polynomial \(x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 83 x^{6} + 48 x^{5} + 525 x^{4} - 1210 x^{3} + 456 x^{2} + 168 x + 784\)

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.