Genus 2 curves in isogeny class 139968.b
Label | Equation |
---|---|
139968.b.839808.1 | \(y^2 + x^3y = x^3 + 3\) |
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
\(\mathrm{ST} =\) $J(E_6)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.2.45349632.2 with defining polynomial:
\(x^{6} - 12\)
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
\(g_4 = 72 b^{5} + 540 b^{2}\)
\(g_6 = -9072 b^{3} - 45360\)
Conductor norm: 9
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -72 b^{5} + 540 b^{2}\)
\(g_6 = 9072 b^{3} - 45360\)
Conductor norm: 9
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 \(x^{12} - 3 x^{10} - 30 x^{8} + 77 x^{6} + 510 x^{4} + 537 x^{2} + 169\)
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.