Given the Dedekind eta function $\eta(\tau)$, define,

$$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$

$$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$

$$\quad\gamma(\tau) =\frac{\alpha(\tau)}{\beta(\tau)} =\frac{\sqrt2\,\eta(4\tau)}{\eta(\tau)}$$

Note that $\alpha^8(\tau)=\lambda(2\tau)$ with *modular lambda function* $\lambda(\tau)$. We have the well-known,

$$\alpha^8(\tau)+\beta^8(\tau) = 1\tag1$$

as well as the nice,

$$\frac{1}{\beta^2(\tau)}-\beta^2(\tau) =\left(2^{1/4}\,\gamma(2\tau)\right)^4\tag2$$

$$\frac{1}{\alpha^2(2\tau)}-\alpha^2(2\tau) =\left(\frac{2^{1/4}}{\gamma(\tau)}\right)^4\tag3$$

and,

$$\frac{1}{\alpha^2(2\tau)}+\alpha^2(2\tau) =\left(\frac{2^{1/4}}{\alpha(\tau)}\right)^4\tag4$$

$$\frac{1}{\beta^2(\tau)}+\beta^2(\tau) =\left(\frac{2^{1/4}}{\beta(2\tau)}\right)^4\tag5$$

As eta functions (not as quotients $\alpha$ and $\beta$), these 5 are in Somos' database (as *t4_24_48, t8_12_24, t8_12_48, t8_18_60a, t8_18_60b*). After some algebraic manipulation, I realized these can be expressed in more aesthetic forms. (Of course, it is easy to transform these to the form $x^2+y^2=1$.)

**Questions:**

- For the next step, can we express $$\frac{1}{\beta^4(\tau)}+\beta^4(\tau) =t_1^2$$ $$\frac{1}{\alpha^4(2\tau)}+\alpha^4(2\tau) =t_2^2$$ where the $t_i$ are eta quotients?
- Are there are other eta quotient parameterizations
**strictly of form**, $$\left(m_1\prod\eta(a_1\tau)^{c_1}\,\eta(a_2\tau)^{c_2}\dots\right)^2 + \left(m_2\prod\eta(b_1\tau)^{d_1}\,\eta(b_2\tau)^{d_2}\dots\right)^2 = 1$$ where $a_i, b_i, c_i, d_i$ are**integers**(the exponents $c_i,d_i$ may be negative) and $m_i$ are algebraic similar to the above 5?