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From Notices of the AMS

The Quintic, the Icosahedron, and Elliptic Curves

The Brioschi parameter B(z) on S2. The absolute value maps to the brightness, while the argument maps to the hue. Image of light blue sphere with five-pointed flower images with black center and purple, pink, red, yellow, and green colored petals.

by Bruce Bartlett
Communicated by William McCallum

There is a remarkable relationship between the roots of a quintic polynomial, the icosahedron, and elliptic curves. This discovery is principally due to Felix Klein (1878), but Klein's marvellous book [9] misses a trick or two, and doesn't tell the whole story. The purpose of this article is to present this relationship in a fresh, engaging, and concise way. We will see that there is a direct correspondence between:

  • "Evenly ordered" roots $(x_1, \ldots, x_5)$ of a Brioschi quintic \begin{equation} X^5 + 10BX^3 + 45B^2X + B^2 = 0, \label{brioschi_intro} \end{equation}
  • Points on the icosahedron, and
  • Elliptic curves equipped with a primitive basis for their 5-torsion, up to isomorphism.

Moreover, this correspondence gives us a very efficient direct method to actually calculate the roots of a general quintic! For this, we'll need some tools both new and old, such as Cremona and Thongjunthug's complex arithmetic geometric mean [3], and the Rogers-Ramanujan continued fraction [5][12]. These tools are not found in Klein's book, as they had not been invented yet!

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