Genus 2 curves in isogeny class 12321.a
Label | Equation |
---|---|
12321.a.36963.1 | \(y^2 + (x^3 + x + 1)y = x^5 + 3x^4 + 4x^3 + 2x^2\) |
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 \(\Q (b) \simeq \) 6.6.69343957.1 with defining polynomial:
\(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)
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
\(g_4 = \frac{27938}{11} b^{5} - \frac{102400}{11} b^{4} + \frac{41669}{11} b^{3} + \frac{90470}{11} b^{2} - \frac{45088}{11} b + \frac{110611}{11}\)
\(g_6 = \frac{32589452}{11} b^{5} - \frac{166685962}{11} b^{4} + \frac{182285865}{11} b^{3} + \frac{203348152}{11} b^{2} - \frac{315712416}{11} b - \frac{19805878}{11}\)
Conductor norm: 729
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 \(K = \Q (a) \simeq \) 6.6.69343957.1 with defining polynomial \(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)
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.