Genus 2 curves in isogeny class 20736.f
Label | Equation |
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20736.f.186624.1 | \(y^2 = x^5 - 2x^4 - 9x^3 - 7x^2 - x\) |
L-function data
Analytic rank: | \(0\) | ||||||||||||||||||||
Mordell-Weil rank: | \(0\) | ||||||||||||||||||||
Bad L-factors: |
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Good L-factors: |
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See L-function page for more information |
Sato-Tate group
\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{36})^+\) with defining polynomial:
\(x^{6} - 6 x^{4} + 9 x^{2} - 3\)
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
\(g_4 = 1174176 b^{5} + 800928 b^{4} - 6485184 b^{3} - 4420656 b^{2} + 7503840 b + 5128272\)
\(g_6 = -586185984 b^{5} - 401008320 b^{4} + 3245158080 b^{3} + 2221502112 b^{2} - 3763179648 b - 2574851328\)
Conductor norm: 4096
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{36})^+\) with defining polynomial \(x^{6} - 6 x^{4} + 9 x^{2} - 3\)
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.