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Properties

Label 102400.b
Conductor $102400$
Sato-Tate group $J(E_2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 102400.b

Label Equation
102400.b.102400.1 \(y^2 = x^5 - x^3 - x\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(5\)\( 1 + T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 4 T^{2} + 9 T^{4}\)
\(7\)\( 1 + 12 T^{2} + 49 T^{4}\)
\(11\)\( 1 + 14 T^{2} + 121 T^{4}\)
\(13\)\( 1 - 22 T^{2} + 169 T^{4}\)
\(17\)\( ( 1 + 2 T + 17 T^{2} )^{2}\)
\(19\)\( 1 + 6 T^{2} + 361 T^{4}\)
\(23\)\( 1 + 44 T^{2} + 529 T^{4}\)
\(29\)\( 1 + 6 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_2)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial:
  \(x^{4} + 1\)

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
  \(g_4 = 160 b^{2} - 48\)
  \(g_6 = 1792 b^{3} - 1152 b\)
   Conductor norm: 160000

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)

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.