Genus 2 curves in isogeny class 123201.a
Label | Equation |
---|---|
123201.a.123201.1 | \(y^2 + (x^3 + 1)y = x^3 + 3\) |
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
\(\mathrm{ST} =\) $J(E_6)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.2.270672597.1 with defining polynomial:
\(x^{6} - 13\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 108 b^{5} + 585 b^{2}\)
\(g_6 = -14742 b^{3} - 56511\)
Conductor norm: 729
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -108 b^{5} + 585 b^{2}\)
\(g_6 = 14742 b^{3} - 56511\)
Conductor norm: 729
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 \(K = \Q (a)\) with defining polynomial \(x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 115 x^{6} - 48 x^{5} + 285 x^{4} - 570 x^{3} + 216 x^{2} + 72 x + 144\)
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.