Genus 2 curves in isogeny class 391876.b
Label | Equation |
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391876.b.783752.1 | \(y^2 + (x^3 + x + 1)y = -x^6 - 2x^5 + 3x^3 - x^2 - 3x - 1\) |
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 \(\Q (b) \simeq \) 6.6.3004150512793.1 with defining polynomial:
\(x^{6} - x^{5} - 130 x^{4} + 481 x^{3} + 2170 x^{2} - 6196 x - 15304\)
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
\(g_4 = \frac{565236}{269} b^{5} - \frac{6779127}{1076} b^{4} - \frac{280105729}{1076} b^{3} + \frac{824114745}{538} b^{2} + \frac{1585191241}{1076} b - \frac{8578763427}{538}\)
\(g_6 = -\frac{631755361055}{2152} b^{5} + \frac{1903087063619}{2152} b^{4} + \frac{39149248059607}{1076} b^{3} - \frac{461440519641675}{2152} b^{2} - \frac{221159457020257}{1076} b + \frac{600559022448280}{269}\)
Conductor norm: 64
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 \(K = \Q (a) \simeq \) 6.6.3004150512793.1 with defining polynomial \(x^{6} - x^{5} - 130 x^{4} + 481 x^{3} + 2170 x^{2} - 6196 x - 15304\)
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.