Genus 2 isogeny class 537289.a

• Label: 537289.a
• Conductor: $537289$
• 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 537289.a

Label	Equation
537289.a.537289.1	\(y^2 + (x^2 + x)y = x^5 - 9x^4 + 12x^3 - 4x^2 - x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(733\)	\( 1 + 50 T + 733 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 - T^{2} + 4 T^{4}\)
\(3\)	\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)	\( ( 1 - 5 T^{2} )^{2}\)
\(7\)	\( ( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )\)
\(11\)	\( 1 - 10 T^{2} + 121 T^{4}\)
\(13\)	\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)	\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}\)
\(19\)	\( ( 1 - 4 T + 19 T^{2} )^{2}\)
\(23\)	\( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4}\)
\(29\)	\( ( 1 - 29 T^{2} )^{2}\)
$\cdots$	$\cdots$

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 \(\Q (b)\) with defining polynomial:
  \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)

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
  \(g_4 = -\frac{8658775}{12288} b^{5} + \frac{3452205}{2048} b^{4} + \frac{870793155}{4096} b^{3} + \frac{23982982945}{12288} b^{2} + \frac{41041673755}{6144} b + \frac{32513544835}{4096}\)
  \(g_6 = -\frac{5246678395}{36864} b^{5} + \frac{50475704531}{147456} b^{4} + \frac{2110058027021}{49152} b^{3} + \frac{907064455213}{2304} b^{2} + \frac{66151883495221}{49152} b + \frac{235489792062851}{147456}\)
   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 \(K = \Q (a)\) with defining polynomial \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)

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.