The name of the equianharmonic curve

Question

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.

Answer 1

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.

If you read German, you might want to know how Wiener described this when he explained the concept in 1901:

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