Genus 2 isogeny class 6400.f

• Label: 6400.f
• Conductor: $6400$
• Sato-Tate group: $E_4$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Genus 2 curves in isogeny class 6400.f

Label	Equation
6400.f.64000.1	\(y^2 + x^3y = -2x^4 - 3x^3 + x^2 + 6x + 4\)

L-function data

Analytic rank:

\(2\)  (upper bound)

Mordell-Weil rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T + 2 T^{2}\)
\(5\)	\( 1 + 4 T + 5 T^{2}\)

Good L-factors:

Prime	L-Factor
\(3\)	\( ( 1 + 2 T + 3 T^{2} )^{2}\)
\(7\)	\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}\)
\(11\)	\( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4}\)
\(13\)	\( 1 - 22 T^{2} + 169 T^{4}\)
\(17\)	\( 1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}\)
\(19\)	\( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}\)
\(29\)	\( ( 1 + 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_4$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.4.256000.1 with defining polynomial:
  \(x^{4} - 20 x^{2} + 10\)

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
  \(g_4 = \frac{1314223139840}{9464869683} b^{3} + \frac{5861441561600}{9464869683} b^{2} - \frac{276644126720}{9464869683} b - \frac{2349793825280}{9464869683}\)
  \(g_6 = -\frac{7920508872718008320}{531632265224427} b^{3} - \frac{35011457079460659200}{531632265224427} b^{2} + \frac{3787394125533593600}{531632265224427} b + \frac{17636598900106035200}{531632265224427}\)
   Conductor norm: 1

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 4.4.256000.1 with defining polynomial \(x^{4} - 20 x^{2} + 10\)

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.