Genus 2 isogeny class 52441.a

• Label: 52441.a
• Conductor: $52441$
• 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 52441.a

Label	Equation
52441.a.52441.1	\(y^2 + (x^2 + x)y = x^5 + 5x^4 + 5x^3 - x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(229\)	\( 1 - 22 T + 229 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 - T^{2} + 4 T^{4}\)
\(3\)	\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)	\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}\)
\(7\)	\( ( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\( ( 1 + 11 T^{2} )^{2}\)
\(13\)	\( ( 1 - 13 T^{2} )^{2}\)
\(17\)	\( ( 1 - 6 T + 17 T^{2} )^{2}\)
\(19\)	\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}\)
\(23\)	\( 1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4}\)
\(29\)	\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.6.629763392149.1 with defining polynomial:
  \(x^{6} - x^{5} - 95 x^{4} + 530 x^{3} - 925 x^{2} + 367 x + 187\)

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
  \(g_4 = \frac{780877}{768} b^{5} - \frac{524729}{384} b^{4} - \frac{24641067}{256} b^{3} + \frac{439363283}{768} b^{2} - \frac{216302341}{192} b + \frac{547887239}{768}\)
  \(g_6 = \frac{788918053}{1152} b^{5} - \frac{2005700851}{2304} b^{4} - \frac{49774062107}{768} b^{3} + \frac{219170907571}{576} b^{2} - \frac{566362435661}{768} b + \frac{1048742008325}{2304}\)
   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 \(K = \Q (a) \simeq \) 6.6.629763392149.1 with defining polynomial \(x^{6} - x^{5} - 95 x^{4} + 530 x^{3} - 925 x^{2} + 367 x + 187\)

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.