Properties

Label 2457.b
Conductor $2457$
Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 2457.b

Label Equation
2457.b.199017.1 \(y^2 + (x^2 + x)y = -x^5 + 4x^4 + x^3 - 13x^2 - 9x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( 1 + T\)
\(7\)\( ( 1 + T )( 1 + 7 T^{2} )\)
\(13\)\( ( 1 - T )( 1 + 2 T + 13 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 - T^{2} + 4 T^{4}\)
\(5\)\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )\)
\(11\)\( 1 - 10 T^{2} + 121 T^{4}\)
\(17\)\( ( 1 - 17 T^{2} )^{2}\)
\(19\)\( ( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( 1 + 14 T^{2} + 529 T^{4}\)
\(29\)\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial:
  \(x^{2} - x + 1\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.0.3.1-273.4-a
  Elliptic curve isogeny class 2.0.3.1-273.1-a

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Endomorphism algebra over \(\overline{\Q}\):

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

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.