Genus 2 isogeny class 72900.a

• Label: 72900.a
• Conductor: $72900$
• 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 72900.a

Label	Equation
72900.a.291600.1	\(y^2 + y = x^6 - 2x^3 + 1\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

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

Good L-factors:

Prime	L-Factor
\(7\)	\( 1 + 7 T^{2} + 49 T^{4}\)
\(11\)	\( ( 1 - 3 T + 11 T^{2} )( 1 + 3 T + 11 T^{2} )\)
\(13\)	\( ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )\)
\(17\)	\( ( 1 + 17 T^{2} )^{2}\)
\(19\)	\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 19 T^{2} + 529 T^{4}\)
\(29\)	\( ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.2.450000.1 with defining polynomial:
  \(x^{6} - x^{5} - 5 x^{3} - x + 1\)

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
  \(g_4 = 192 b^{5} - 690 b^{4} + 690 b^{3} - 750 b^{2} + 2280 b - 672\)
  \(g_6 = -20160 b^{5} + 10080 b^{4} + 20160 b^{3} + 80640 b^{2} + 40320 b - 39060\)
   Conductor norm: 4782969
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -192 b^{5} - 210 b^{4} + 210 b^{3} + 1650 b^{2} + 1320 b + 672\)
  \(g_6 = 20160 b^{5} - 10080 b^{4} - 20160 b^{3} - 80640 b^{2} - 40320 b - 28980\)
   Conductor norm: 4782969

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 \(x^{12} - x^{11} + x^{10} + 10 x^{9} - 5 x^{8} - x^{7} + 26 x^{6} - x^{5} - 5 x^{4} + 10 x^{3} + x^{2} - x + 1\)

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.