Genus 2 curves in isogeny class 262144.d
Label | Equation |
---|---|
262144.d.524288.1 | \(y^2 = x^5 - x^4 + 4x^3 - 8x^2 + 5x - 1\) |
262144.d.524288.2 | \(y^2 = x^5 + x^4 + 4x^3 + 8x^2 + 5x + 1\) |
L-function data
Analytic rank: | \(1\) | ||||||||||||||||||||||
Mordell-Weil rank: | \(1\) | ||||||||||||||||||||||
Bad L-factors: |
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Good L-factors: |
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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
Simple over \(\overline{\Q}\)
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 \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | the quaternion algebra over \(\Q\) of discriminant 6 |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.