Genus 2 isogeny class 8281.c

• Label: 8281.c
• Conductor: $8281$
• 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 8281.c

Label	Equation
8281.c.405769.1	\(y^2 + (x^2 + x)y = x^5 + 8x^4 + 11x^3 + 3x^2 - x\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(7\)	\( 1 + 4 T + 7 T^{2}\)
\(13\)	\( 1 + 2 T + 13 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 - T^{2} + 4 T^{4}\)
\(3\)	\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}\)
\(5\)	\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}\)
\(11\)	\( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}\)
\(17\)	\( ( 1 - 6 T + 17 T^{2} )^{2}\)
\(19\)	\( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4}\)
\(23\)	\( ( 1 + 23 T^{2} )^{2}\)
\(29\)	\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

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 \(\Q (b) \simeq \) 6.6.891474493.1 with defining polynomial:
  \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391\)

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
  \(g_4 = \frac{6900061}{184} b^{5} + \frac{2308663}{368} b^{4} - \frac{213373265}{184} b^{3} - \frac{110121753}{92} b^{2} + \frac{189026334}{23} b + \frac{212819299}{16}\)
  \(g_6 = -\frac{14940873857}{1472} b^{5} - \frac{1373744645}{736} b^{4} + \frac{7227770823}{23} b^{3} + \frac{241586544017}{736} b^{2} - \frac{3281620049799}{1472} b - \frac{14496977859}{4}\)
   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 \(K = \Q (a) \simeq \) 6.6.891474493.1 with defining polynomial \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391\)

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.