# Klein's Quartic

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Klein Quartic Talk
Greg Egan's Equation Page
Greg Egan's Curve Page
John Baez's Page

## Resources

The Eightfold Way: The Beauty of Klein’s Quartic Curve, edited Silvio Levy
Noam D. Elkies. The Klein Quartic in Number Theory. In The Eightfold Way, vol 35. MSRI Publications, 1998.

## Modular Curve X(7)

### Symmetry Group

The symmetry group (orientation preserving automorphsims) is of size 168. It is isomorphic to $G\cong {\mathrm{PSL}}_{2}\left({\mathbb{F}}_{7}\right)\cong G{L}_{3}\left({\mathbb{F}}_{2}\right)$ The character table is given by $\overline{)\begin{array}{ccccccc}c& 1A& 2A& 3A& 4A& 7A& 7B\\ \mathrm{#}c& 1& 21& 56& 42& 24& 24\\ {\chi }_{1}& 1& 1& 1& 1& 1& 1\\ {\chi }_{3}& 3& -1& 0& 1& \alpha & \overline{\alpha }\\ {\overline{\chi }}_{3}& 3& -1& 0& 1& \overline{\alpha }& \alpha \\ {\chi }_{6}& 6& 2& 0& 0& -1& -1\\ {\chi }_{7}& 7& -1& 1& -1& 0& 0\\ {\chi }_{8}& 8& 0& -1& 0& 1& 1\end{array}}$

### Klein Model

The ring of invariant polynomial of the three dimensional representation is generated by ${\mathrm{\Phi }}_{4}:={X}^{3}Y+{Y}^{3}Z+{Z}^{3}X$ ${\mathrm{\Phi }}_{6}:=-\frac{1}{54}H\left({\mathrm{\Phi }}_{4}\right)=X{Y}^{5}+Y{Z}^{5}+Z{X}^{5}-5{X}^{2}{Y}^{2}{Z}^{2}$ ${\mathrm{\Phi }}_{14}=\frac{1}{9}|\begin{array}{cccc}{\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }{X}^{2}& {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }X\mathrm{\partial }Y& {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }X\mathrm{\partial }Z& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }X\\ {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }Y\mathrm{\partial }X& {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }{Y}^{2}& {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }Y\mathrm{\partial }Z& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }Y\\ {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }Z\mathrm{\partial }X& {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }Z\mathrm{\partial }Y& {\mathrm{\partial }}^{2}{\mathrm{\Phi }}_{4}/\mathrm{\partial }{Z}^{2}& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }Z\\ \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }X& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }Y& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }Z& 0\end{array}|$ ${\mathrm{\Phi }}_{21}=\frac{\mathrm{\partial }\left({\mathrm{\Phi }}_{4},{\mathrm{\Phi }}_{6},{\mathrm{\Phi }}_{14}\right)}{14\mathrm{\partial }\left(X,Y,Z\right)}=\frac{1}{14}|\begin{array}{lll}\mathrm{\partial }{\mathrm{\Phi }}_{4}/\mathrm{\partial }X& \mathrm{\partial }{\mathrm{\Phi }}_{4}/\mathrm{\partial }Y& \mathrm{\partial }{\mathrm{\Phi }}_{4}/\mathrm{\partial }Z\\ \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }X& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }Y& \mathrm{\partial }{\mathrm{\Phi }}_{6}/\mathrm{\partial }Z\\ \mathrm{\partial }{\mathrm{\Phi }}_{14}/\mathrm{\partial }X& \mathrm{\partial }{\mathrm{\Phi }}_{14}/\mathrm{\partial }Y& \mathrm{\partial }{\mathrm{\Phi }}_{14}/\mathrm{\partial }Z\end{array}|$

### Ramanujan Theta Series on X(7)

The map from the modular curve X(7) to the Klein Quartic is given by (a: b: c) where $\begin{array}{rl}& \mathrm{a}\left(z\right)=-\sum _{n\in \mathbb{Z}}\left(-1{\right)}^{n}{q}^{\left(14n+5{\right)}^{2}/56}\\ & \text{}\mathrm{b}\left(z\right)=\sum _{n\in \mathbb{Z}}\left(-1{\right)}^{n}{q}^{\left(14n+3{\right)}^{2}/56}\\ & \mathrm{c}\left(z\right)=\sum _{n\in \mathbb{Z}}\left(-1{\right)}^{n}{q}^{\left(14n+1{\right)}^{2}/56}\end{array}$