Genus 2 curves in isogeny class 17689.b
Label | Equation |
---|---|
17689.b.17689.1 | \(y^2 + (x^3 + x + 1)y = -3x^4 - 3x^3 + x^2 + x\) |
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 \(\Q (b) \simeq \) 6.6.41615795893.2 with defining polynomial:
\(x^{6} - x^{5} - 55 x^{4} + 160 x^{3} + 246 x^{2} - 1107 x + 729\)
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
\(g_4 = -\frac{29571725}{89424} b^{5} + \frac{12875}{552} b^{4} + \frac{1628558075}{89424} b^{3} - \frac{268027405}{7452} b^{2} - \frac{95055545}{828} b + \frac{5963540}{23}\)
\(g_6 = \frac{391613779963}{3219264} b^{5} - \frac{1529536043}{178848} b^{4} - \frac{21563971628059}{3219264} b^{3} + \frac{7102017353893}{536544} b^{2} + \frac{2517427711897}{59616} b - \frac{421325658103}{4416}\)
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 \(K = \Q (a) \simeq \) 6.6.41615795893.2 with defining polynomial \(x^{6} - x^{5} - 55 x^{4} + 160 x^{3} + 246 x^{2} - 1107 x + 729\)
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.