Genus 2 curves in isogeny class 589824.c
Label | Equation |
---|---|
589824.c.589824.1 | \(y^2 = x^5 - 2x^4 - 4x^3 + 8x^2 + x - 2\) |
589824.c.589824.2 | \(y^2 = x^5 + 2x^4 - 4x^3 - 8x^2 + x + 2\) |
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
\(\mathrm{ST} =\) $J(E_2)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial:
\(x^{4} + 1\)
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
\(g_4 = -1024 b^{3} - 3072 b^{2} - 3584 b - 1920\)
\(g_6 = 329728 b^{3} + 100352 b^{2} - 186368 b - 368640\)
Conductor norm: 5308416
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)
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.