
> 
> **I. Klein**
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> 
> 


In "*[On the Order-Seven Transformation of Elliptic Functions](http://library.msri.org/books/Book35/files/klein.pdf)*" (pp. 287-331), he discusses in p. 298 what we now call the *Klein quartic*,


$$\lambda^3\mu+\mu^3\nu+\nu^3\lambda= 0\tag1$$


and in p. 313 introduces what we can call the *Klein septic resolvent*,


$$z^7-2^2\cdot7^2\big(7\mp\sqrt{-7}\big)z^4+2^5\cdot 7^4\big(5\mp\sqrt{-7}\big)\color{red}z\\ \mp 2^9\cdot3\cdot7^3\sqrt{-7}\frac{g\_2}{\sqrt[3]\Delta}=0\tag2$$


(where the red linear $\color{red}z$ is missing in the paper and I assume is a typo). In the same page, he says the roots of $(2)$ in terms of $\lambda,\mu, \nu$ are,


$$z =\frac{\pm 2\sqrt{-7}\Big(P\_1+\frac{-1\mp\sqrt{-7}}{2}P\_2\Big)}{\sqrt[3]\nabla}\tag3$$


where,


$$P\_1 = \gamma^{2x}\lambda^2+\gamma^{x}\mu^2+\gamma^{4x}\nu^2\\
P\_2 =\gamma^{6x}\mu\nu+\gamma^{3x}\nu\lambda+\gamma^{5x}\lambda\mu$$


with $\gamma= e^{2\pi i/7}$ as first mentioned in p. 313.



> 
> **II. Ramanujan**
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> 
> 


Unbeknownst to Klein (d. 1925), it turns out Ramanujan (d. 1920) found an elegant parameterization to $(1)$. Define the [Ramanujan theta function](https://en.wikipedia.org/wiki/Ramanujan_theta_function),


$$f(a,b) = \sum\_{n=-\infty}^\infty a^{n(n+1)/2}\, b^{n(n-1)/2}$$


then,


$$\lambda = q^{1/56}f(-q^3,-q^4)\\ \mu= \color{red}{-}q^{25/56}f(-q,-q^6)\\ \nu= q^{9/56}f(-q^2,-q^5)\\$$


where $q=e^{2\pi i\tau}$.



> 
> **III. Question**
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> 


I tried to implement this in *Mathematica*. Unfortunately, I couldn't get $(3)$ to be a root of the septic $(2)$. 


**Q:** Was I wrong in assuming that ***any*** parametrization to $(1)$ would do? Or is there some typo or confusion of variables in the paper that screwed up my implementation?



Can we use Ramanujan's parameterization of Klein's quartic to solve Klein's septic?
There seems to be a problem with Klein's septic equation $(2)$ combined with the purported roots in $(3)$. Let $\,k\,$ be any integer. Define
$$ P\_1(k) := \gamma^{k}\mu^2 + \gamma^{2k}\lambda^2 + \gamma^{4k}\nu^2 \tag{1} $$
and 
$$ P\_2(k) := \gamma^{3k}\lambda\nu + \gamma^{6k}\mu\nu +
 \gamma^{5k}\lambda\mu. \tag{2} $$
Let
$$ \,a := 2\sqrt{-7}/\eta(\tau)^2 \; \textrm{ and } \; b := -(7+\sqrt{-7})/\eta(\tau)^2. \tag{3} $$
 Define the roots of $(2)$ as
$$ r\_k := a\,P\_1(k) + b\,P\_2(k). \tag{4} $$
 The polynomial
$\, P(z) := (z - r\_1)(z - r\_2)\cdots (z - r\_7)\,$ expands to 
$$ P(z) = z^7 \!-\! 2^2\! \cdot\! 7^2\, (7+\sqrt{-7})\, z^4 \!+\! 2^5\!\cdot \!7^4\,
 (5+\sqrt{-7})\,z \!+\! 2^7\! \cdot\! 7^3 \sqrt{-7} \frac{g\_2(\tau)}{\eta(\tau)^8}. \tag{5}$$
This was for values of $\,\lambda,\mu,\nu\,$ as suggested by Tito Piezas III.


**P.S.** Note that
$\, \sqrt[3]{\Delta} = \eta(\tau)^8$ is the denominator of the constant term of $P(z).$ Also note that $\, 1728 J(\tau) = j(\tau) = g\_2(\tau)^3/\eta(q)^{24}\,$ where $\,J(\tau)\,$ is [Klein's invariant](http://mathworld.wolfram.com/KleinsAbsoluteInvariant.html). Thus using cube roots
$\, 12\sqrt[3]{J(\tau)} = g\_2(\tau)/\eta(\tau)^8\,$ and the constant term
can be written as
$\, 2^9\!\cdot 3 \cdot\! 7^3 \sqrt{-7} \sqrt[3]{J(\tau)} \,$ which is closer to Klein's version. In fact, just before Klein's equation $(49)$ (Tito's equation $(2)$) Klein writes $\ J = g\_2^3/\Delta.$


**P.P.S.** I see that Tito has used a set of roots $\,y\_k\,$ differing by a common factor from $\,z\_k\,$ to simplify the septic. Perhaps Klein would have used that version, but he preferred $\,J(\tau)\,$ instead of $\,j(\tau).$


