Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On parameterizations of cyclic $N$-isogenies and strict $K$-curves lying
  above rational points of $Y_0^+(N)$   Elliptic $K$-curves are elliptic curves defined over some field extension
$L/K$ that are isogenous to all of their Galois conjugates. We present a new
result on $K$-curves $E$ that are given by a $K$-rational orbit $\{\tau,
-1/N\tau\}$ of the Fricke involution on $Y_0(N)$, giving a simple Diophantine
condition on the extension $L/K$ that determines which twists of $E$ allow the
isogeny between Galois conjugates to be defined over $L$. To support and
illustrate this result, we also discuss parameterizations of cyclic
$N$-isogenies corresponding to points on modular curves $X_0(N)$ of genus $0$.
These modular curves admit parameterizations in terms of a distinguished
Hauptmodul. We provide an exposition on the derivation of these Hauptmoduln as
products of the Dedekind eta function based on the approach of Ligozat. As an
application, we provide a complete tabulation of explicit formulas for the
coefficients of cyclic $N$-isogenous curves in terms of the Hauptmodul for all
$N$ such that $X_0(N)$ has genus $0$. We also include an abbreviated table of
rational functions for the $j$-invariant in terms of Hauptmoduln, and we
discuss a classical application of these expressions to finding special values
of the $j$-invariant at CM points.

23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci