Genus 2 curves in isogeny class 12544.g
Label | Equation |
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12544.g.175616.1 | \(y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2\) |
L-function data
Analytic rank: | \(1\) | ||||||||||||||||||||
Mordell-Weil rank: | \(1\) | ||||||||||||||||||||
Bad L-factors: |
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Good L-factors: |
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See L-function page for more information |
Sato-Tate group
\(\mathrm{ST} =\) $J(E_4)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 4.0.2744.1 with defining polynomial:
\(x^{4} - x^{3} + 3 x^{2} + 3 x + 2\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -56 b^{3} + 112 b^{2} + 280 b + 112\)
\(g_6 = -812 b^{3} + 8596 b^{2} + 7700 b + 5124\)
Conductor norm: 1024
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -280 b^{3} + 560 b^{2} - 1288 b + 112\)
\(g_6 = -16338 b^{3} + 25704 b^{2} - 59150 b - 22764\)
Conductor norm: 1024
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism algebra over \(\Q\):
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.481890304.3 with defining polynomial \(x^{8} - 2 x^{7} - 3 x^{6} + 8 x^{5} + x^{4} - 2 x^{3} + 23 x^{2} - 12 x + 2\)
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.