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Properties

Label 102400.e
Conductor $102400$
Sato-Tate group $J(E_1)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 102400.e

Label Equation
102400.e.102400.1 \(y^2 = x^5 + 3x^3 + x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(5\)\( ( 1 - T )^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(7\)\( ( 1 - 2 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)\( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )^{2}\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )^{2}\)
\(19\)\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)
\(29\)\( ( 1 - 2 T + 29 T^{2} )^{2}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_1)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 320.e
  Elliptic curve isogeny class 320.b

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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.