Abstract
Since the genus of the modular curve X_1 (8) = Γ _1 (8)∖\(\mathfrak{h}\)* is zero, we find a field generator j 1,8(z) = θ3(2z)/θ3(4z) (θ3(z) := ∑ nɛℤ eπin 2z) such that the function field over X 1(8) is ℂ(j 1,8). We apply this modular function j 1,8 to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j 1,8) of ℂ(j 1,8).
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Kim, C.H., Koo, J.K. Arithmetic of the Modular Function j1,8. The Ramanujan Journal 4, 317–338 (2000). https://doi.org/10.1023/A:1009857205327
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DOI: https://doi.org/10.1023/A:1009857205327