Genus 2 curves in isogeny class 3969.b
Label | Equation |
---|---|
3969.b.35721.1 | \(y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - 3x^2\) |
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 \(\Q (b) \simeq \) 6.6.330812181.1 with defining polynomial:
\(x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35\)
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
\(g_4 = \frac{29108241}{62500} b^{5} + \frac{81976293}{250000} b^{4} - \frac{1198505889}{125000} b^{3} - \frac{3288044907}{250000} b^{2} + \frac{5106100923}{250000} b + \frac{1196447679}{50000}\)
\(g_6 = -\frac{248712019479}{3125000} b^{5} - \frac{1355533889409}{25000000} b^{4} + \frac{10205353689741}{6250000} b^{3} + \frac{55772501101191}{25000000} b^{2} - \frac{86931059013099}{25000000} b - \frac{2533145646969}{625000}\)
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 \(K = \Q (a) \simeq \) 6.6.330812181.1 with defining polynomial \(x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35\)
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.