Let
N
N
be one of the
38
38
distinct square-free integers such that the arithmetic group
Γ
0
(
N
)
+
\Gamma _0(N)^+
has genus one. We constructed canonical generators
x
N
x_N
and
y
N
y_N
for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of
x
N
x_N
, which we express as a polynomial in
y
N
y_N
with coefficients that are rational functions in
x
N
x_N
. As a corollary, we prove that for any point
e
e
in the upper half-plane which is fixed by an element of
Γ
0
(
N
)
+
\Gamma _0(N)^+
, one can explicitly evaluate
x
N
(
e
)
x_N(e)
and
y
N
(
e
)
y_N(e)
. As it turns out, each value
x
N
(
e
)
x_N(e)
and
y
N
(
e
)
y_N(e)
is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of
x
N
(
τ
)
x_N(\tau )
and
y
N
(
τ
)
y_N(\tau )
at any CM point
τ
\tau
, including elliptic points, for any genus one group
Γ
0
(
N
)
+
\Gamma _0(N)^+
. Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.