In this article we observe nedians and nedian triangles of ratio $\eta$ for a given triangle. The locus of vertices of the nedian triangles for $\eta\in\mathbb{R}$ is found and its correlation with isotomic conjugates of the given triangle is shown. Furthermore, the curve on which lie vertices of a nedian triangle for fixed $\eta$, when we iterate nedian triangles, is found.
In this paper we discuss applications of the theory developed in [21] and [22] in studying certain Galois groups and splitting fields of rational functions in Q (X 0 (N )) using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE.2000 Mathematics Subject Classification. 11F11.
We find plane models for all X 0 (N ), N ≥ 2. We observe a map from the modular curve X 0 (N ) to the projective plane constructed using modular forms of weight 12 for the group Γ 0 (N ); the Ramanujan function ∆, ∆(N •) and the third power of Eisestein series of weight 4, E 3 4 , and prove that this map is birational equivalence for every N ≥ 2. The equation of the model is the minimal polynomial of ∆(N •)/∆ over C(j).
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