Genus 2 curves in isogeny class 301401.a
Label | Equation |
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301401.a.301401.1 | \(y^2 + (x^3 + x + 1)y = -x^6 - x^5 + 3x^4 + 4x^3 - 3x^2 - 4x - 1\) |
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 \(\Q (b) \simeq \) 6.6.5541396330861.1 with defining polynomial:
\(x^{6} - 183 x^{4} - 244 x^{3} + 1647 x^{2} + 549 x - 2745\)
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
\(g_4 = -\frac{448847426817}{507012500} b^{5} - \frac{3717968867361}{2028050000} b^{4} + \frac{40106532132459}{253506250} b^{3} + \frac{1102504970084661}{2028050000} b^{2} - \frac{673019993223549}{2028050000} b - \frac{475445034877491}{405610000}\)
\(g_6 = -\frac{60422321890130337}{50701250000} b^{5} - \frac{500673379794933771}{202805000000} b^{4} + \frac{21595984719044897871}{101402500000} b^{3} + \frac{148446854122001801571}{202805000000} b^{2} - \frac{90546645136674896739}{202805000000} b - \frac{32026060517213501163}{20280500000}\)
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 \(K = \Q (a) \simeq \) 6.6.5541396330861.1 with defining polynomial \(x^{6} - 183 x^{4} - 244 x^{3} + 1647 x^{2} + 549 x - 2745\)
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.