Algorithm for finding full representatives of the orbit space of imaginary quadratic numbers of discriminant $D$ under the modular group

Question

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ Im(z) > 0\}$ be the upper half plane of complex numbers. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s \end{array} \right)$ be an element of $\Gamma$. Let $z \in \mathcal{H}$. We write $$\sigma z = \frac{pz + q}{rz + s}$$ It is easy to see that $\sigma z \in \mathcal{H}$ and $\Gamma$ acts on $\mathcal{H}$ from left.

Let $\alpha \in \mathbb{C}$ be an algebraic number. If the minimal polynomial of $\alpha$ over $\mathbb{Q}$ has degree $2$, we say $\alpha$ is a quadratic number. There exists the unique polynomial $ax^2 + bx + c \in \mathbb{Z}[x]$ such that $a > 0$ and gcd$(a, b, c) = 1$. $D = b^2 - 4ac$ is called the discriminant of $\alpha$. Since $D \equiv b^2$ (mod $4$), $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Conversly suppose $D$ is a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Then there exists a quadratic number $\alpha$ whose discriminant is $D$.

Let $D$ be a negative non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). We denote by $\mathcal{H}(D)$ the set of quadratic numbers of discriminant $D$ in $\mathcal{H}$. By this question, $\mathcal{H}(D)$ is $\Gamma$-invariant.

My question Is there algorithm for finding full representatives of the orbit space $\mathcal{H}(D)/\Gamma$? If yes, what is it?

Remark My motivation for the above question came from this and this. This is a closely related question.

Answer 1

We use the notation of this question. Let $D$ be a negative non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). We denote the set of positive definite primitive binary quadratic forms of discriminant $D$ by $\mathfrak{F}^+_0(D)$. By this question, $\mathfrak{F}^+_0(D)$ is $\Gamma$-invariant. We denote the set of $\Gamma$-orbits on $\mathfrak{F}^+_0(D)$ by $\mathfrak{F}^+_0(D)/\Gamma$.

Let $f = ax^2 + bxy + cy^2 \in \mathfrak{F}^+_0(D)$. We denote $\phi(f) = (-b + \sqrt{D})/2a$, where $\sqrt{D} = i\sqrt{|D|}$. It is clear that $\phi(f) \in \mathcal{H}(D)$. Hence we get a map $\phi\colon \mathfrak{F}^+_0(D) \rightarrow \mathcal{H}(D)$. By this question, $\phi$ is a bijection and induces a bijection $\mathfrak{F}^+_0(D)/\Gamma \rightarrow \mathcal{H}(D)/\Gamma$. Hence, it suffices to find algorithm to determine full representatives of $\mathfrak{F}^+_0(D)/\Gamma$. For notational convenience, we denote an element $ax^2 + bxy + cy^2$ of $\mathfrak{F}^+_0(D)$ by $(a, b, c)$.

Let $F = \{ z \in \mathcal{H}\ |\ -1/2 \le Re(z) \lt 1/2, |z| \gt 1$ or $|z| = 1$ and $Re(z) \le 0 \}$. $F$ is a fundamental domain of $\mathcal{H}/\Gamma$, i.e. every point of $\mathcal{H}$ is $\Gamma$-equivalent to a unique point of $F$(e.g. Serre's A Course in Arithmetic or Proposition 1 and Proposition 2 of my answer to this question).

Let $\mathcal{F}(D) = \{(a, b, c) \in \mathfrak{F}^+_0(D)\ |\ |b| \le a \le c $ If $|b| = a$ or $a = c$, then $b \ge 0\}$. It is easy to see that $\phi^{-1}(F) = \mathcal{F}(D)$(see Proposition 3 of my answer to this question). Hence it suffices to determine the set $\mathcal{F}(D)$. This is done in my answer to this question.