Properties

Label 301401.a
Conductor $301401$
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 301401.a

Label Equation
301401.a.301401.1 \(y^2 + (x^3 + x + 1)y = -x^6 - x^5 + 3x^4 + 4x^3 - 3x^2 - 4x - 1\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( 1 + 3 T^{2}\)
\(61\)\( 1 + 13 T + 61 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4}\)
\(5\)\( ( 1 - 3 T + 5 T^{2} )^{2}\)
\(7\)\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 + 5 T + 13 T^{2} )^{2}\)
\(17\)\( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)\( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 55 T^{2} + 841 T^{4}\)
$\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 \(\Q (b) \simeq \) 6.6.5541396330861.1 with defining polynomial:
  \(x^{6} - 183 x^{4} - 244 x^{3} + 1647 x^{2} + 549 x - 2745\)

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
  \(g_4 = -\frac{448847426817}{507012500} b^{5} - \frac{3717968867361}{2028050000} b^{4} + \frac{40106532132459}{253506250} b^{3} + \frac{1102504970084661}{2028050000} b^{2} - \frac{673019993223549}{2028050000} b - \frac{475445034877491}{405610000}\)
  \(g_6 = -\frac{60422321890130337}{50701250000} b^{5} - \frac{500673379794933771}{202805000000} b^{4} + \frac{21595984719044897871}{101402500000} b^{3} + \frac{148446854122001801571}{202805000000} b^{2} - \frac{90546645136674896739}{202805000000} b - \frac{32026060517213501163}{20280500000}\)
   Conductor norm: 1

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 \(K = \Q (a) \simeq \) 6.6.5541396330861.1 with defining polynomial \(x^{6} - 183 x^{4} - 244 x^{3} + 1647 x^{2} + 549 x - 2745\)

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.