Genus 2 curves in isogeny class 8281.b
Label | Equation |
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8281.b.405769.1 | \(y^2 + (x^3 + x + 1)y = -3x^5 + 9x^4 - 7x^3 - 2x^2 + x\) |
L-function data
Analytic rank: | \(2\) (upper bound) | ||||||||||||||||||||
Mordell-Weil rank: | \(2\) | ||||||||||||||||||||
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 \(\Q (b) \simeq \) 6.6.891474493.2 with defining polynomial:
\(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 162 x^{2} - 81 x - 27\)
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
\(g_4 = \frac{1574069}{6561} b^{5} + \frac{13576892}{6561} b^{4} - \frac{56218393}{6561} b^{3} - \frac{17230402}{243} b^{2} + \frac{2496578}{243} b + \frac{81063052}{243}\)
\(g_6 = -\frac{655459038871}{531441} b^{5} + \frac{100362384668}{531441} b^{4} + \frac{20455326360833}{531441} b^{3} + \frac{543846776200}{19683} b^{2} - \frac{3528969869152}{19683} b - \frac{1057160135213}{19683}\)
Conductor norm: 1
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 \(K = \Q (a) \simeq \) 6.6.891474493.2 with defining polynomial \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 162 x^{2} - 81 x - 27\)
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.