Properties

Label 454276.a
Conductor $454276$
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

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Genus 2 curves in isogeny class 454276.a

Label Equation
454276.a.908552.1 \(y^2 + (x^3 + x + 1)y = -x^5 - 4x^4 + 5x^2 + 2x\)

L-function data

Analytic rank:\(2\)  (upper bound)
Mordell-Weil rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T + T^{2}\)
\(337\)\( 1 + 29 T + 337 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4}\)
\(5\)\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)\( ( 1 + T + 7 T^{2} )^{2}\)
\(11\)\( ( 1 - 11 T^{2} )^{2}\)
\(13\)\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 14 T^{2} + 289 T^{4}\)
\(19\)\( 1 + 19 T^{2} + 361 T^{4}\)
\(23\)\( 1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4}\)
$\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 \(\Q (b) \simeq \) 6.6.4346598285457.1 with defining polynomial:
  \(x^{6} - x^{5} - 140 x^{4} + 443 x^{3} + 3104 x^{2} - 13642 x + 12979\)

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
  \(g_4 = \frac{2037697}{2631496} b^{5} + \frac{11235267}{10525984} b^{4} - \frac{1101252315}{10525984} b^{3} + \frac{935074463}{10525984} b^{2} + \frac{6512139369}{2631496} b - \frac{39280936183}{10525984}\)
  \(g_6 = -\frac{433245604979}{14441650048} b^{5} - \frac{739062362093}{14441650048} b^{4} + \frac{59159166273009}{14441650048} b^{3} - \frac{16966490471535}{7220825024} b^{2} - \frac{1493197781452289}{14441650048} b + \frac{559909913775949}{3610412512}\)
   Conductor norm: 64

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 \(K = \Q (a) \simeq \) 6.6.4346598285457.1 with defining polynomial \(x^{6} - x^{5} - 140 x^{4} + 443 x^{3} + 3104 x^{2} - 13642 x + 12979\)

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.