**I. Level 7**


In Klein's "[*On the Order-Seven Transformations of Elliptic Functions*](http://library.msri.org/books/Book35/files/klein.pdf)", he gave two *elegant* resolvents of degrees 8 and 7 in pages 306 and 313. Translated to more understandable notation, we have,


$$x^8+14x^6+63x^4+70x^2-7 = x\sqrt{j(\tau)-1728}\tag1$$


$$y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j(\tau)\tag2$$


For general $j$, in Magma notation these are 8T43 = **PGL(2,7)** and 14T16, respectively, both of which have order $2\times168 = 336$ hence generally not solvable in radicals. But I found the octic in $x$ has a nice solution by the septic in $y$. Let $\zeta = e^{2\pi i/7}$, then,


$$x\_k = \frac{\alpha}4\left(\pm\sqrt{y\_1+\alpha^6}\pm\sqrt{y\_2+\alpha^6}\pm\dots\pm\sqrt{y\_7+\alpha^6}\,\right)$$


where $\alpha = \zeta+\zeta^2+\zeta^4 = \frac{-1+\sqrt{-7}}2$, a radical in the character table of [PSL(2,7)](https://en.wikipedia.org/wiki/PSL(2,7)), and signs chosen appropriately. This is a similar method used for the 8T25 octic $x^8-x^7+29x^2+29=0$ [here](https://mathoverflow.net/questions/145145/).


However, if $j$ is the **j-function** like $j\Big(\tfrac{1+\sqrt{-163}}2\Big)$, then the order is smaller and *is now solvable in radicals*.




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**II. Level 13**


The equation $(1)$ involves level 7 eta quotients. But this has a level 13 counterpart, namely,


$$\frac{(x^2 + 6x + 13)(x^6 + 10x^5 + 46x^4 + 108x^3 + 122x^2 + 38x - 1)^2}{x} = j(\tau)-1728$$


Analogous to its younger sibling, this is 14T39 = **PGL(2,13)** with order $24\times7\times13=2184$ hence is generally not solvable in radicals. But the order is smaller if $j$ is the j-function and is solvable.




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**III. Question**


Is there a transitive group of degree $13m$ and order $2184$? The [online databases](https://www.lmfdb.org/knowledge/show/nf.galois_group.name) show there are none for $m=1,2$ but is silent for $m>2$.


**P.S.** The objective is create a formula for the j-function with $13$ as the highest exponent, but with coefficients as algebraic numbers of deg $m$, analogous to formula $(2)$.



Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13?
The smallest $m$ for which $\mathrm{PGL}(2,13)$ acts faithfully and transitively on a set of $13m$ elements is $m=6$, with point stabiliser a dihedral group of order $28$. I think you may be seeing a manifestation of Camille Jordan's theorem that $\mathrm{PSL}(2,p)$ has a subgroup of index $p$ if and only if $p\leqslant 11$.


