# The name of the equianharmonic curve

I have found several references where the elliptic curve $$y^2=x^3-1$$ is called the equianharmonic curve, and, more often, where the half-period of this curve $$\omega_1 = \frac{\Gamma(1/3)^3}{4\pi}$$ is called the equianharmonic constant, the corresponding semi-period for the elliptic curve $$y^2=x^3-x$$ is called the Lemniscate constant.

I think that the anharmonic ratio of four points is another name for the cross-ratio, if we look the action of $$S_4$$ in the cross ratio we see that it is fixed by $$K$$, Klein's four-group, and $$S_4/K \simeq S_3$$ is called the anharmonic group, that can be viewed as the group of Mobius transformations $$\lambda, \frac{1}\lambda, 1-\lambda, \frac{1}{1-\lambda}, \frac{1-\lambda}{\lambda}, \frac{\lambda}{1-\lambda}$$ This group has three especial orbits with less than 6 elements: $$\{0,1,\infty\}$$, $$\{-1,\tfrac{1}{2}, 2\}$$ called the harmonic case, and $$\{\pm e^{2\pi i/6}\}$$ which might be called the equianharmonic case but I am bot sure.

My question is why is it called equianharmonic curve/constant? I expect it to have a relationship with the group but I can't figure out which. If so, does the harmonic case have a similar relationship with some other elliptic curve ($$y^2=x^3-x$$ for instance)?

Thank you and sorry if it is a silly question.

• Maybe the idea is that it's not just "anharmonic" (not harmonic) but as far as possible from the "harmonic" case $-$ and the distance is the same (thus "equi-") in each of the three directions that you could deform the tetrahedrally symmetric curve to reach the dihedrally symmetric "harmonic" curve. Feb 27, 2021 at 18:33

The name refers to the concept of an anharmonic ratio, or cross-ratio. Four points $$A,B,C,D$$ are called equianharmonic if their cross-ratio is a cube root of 1. In that case the 6 cross-ratios obtained by combining the four points in all possible ways are the same ("equal anharmonic ratio" = "equianharmonic").
The curve $$y^2=x^3-1=(x-p_1)(x-p_2)(x-p_3)$$ has $$p_1=1$$, $$p_2=e^{i2\pi/3}$$, $$p_3=e^{i4\pi/3}$$ equal to the three cube roots of 1. The four intersection points of a line with the four tangents, drawn from any point on the curve, are equianharmonic.