Genus 2 curves in isogeny class 50000.a
Label | Equation |
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50000.a.200000.1 | \(y^2 + y = 2x^5\) |
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
\(\mathrm{ST} =\) $F_{ac}$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{U}(1)\)
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{5})\) with defining polynomial \(x^{4} - x^{3} + x^{2} - x + 1\)
Endomorphism algebra over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q(\zeta_{5})\) (CM) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\C \times \C\) |
More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.