## Torsion Twirl

by Shelly Manber

Answer: THE TWIST

The eight equations at the bottom are all equations of elliptic curves. The eight videos each give two pieces of
information: a point on an elliptic curve with integer coefficients given by the number of turns the `x` and `y` dancers
do, counterclockwise for positive and clockwise for negative, and a unique ballet turn whose name is enumerated
by the dashes below the video.

Each of the points in the videos are points on one of the curves, which gives the matching between equation
and video. Furthermore, each of the points is a torsion point on its given elliptic curve, *i.e.* a point of finite order
under the elliptic curve group law. In other words, for each of these points, if you add the point to itself some
number of times under the elliptic curve group law, you will get back to the identity. The smallest number of
times a point adds to itself to give the identity is called the order of the point. If you find the order of each of
these torsion points and index into the name of the ballet turn, you will get the answer phrase: THE
TWIST.

Something to note: the fifth video gives the point (0, 1) which is supposed to be a representative of the
projective point (0 : 1 : 0), *i.e.* the identity of the elliptic curve as a group and the unique point of order one. This
is matched with
`y`^{2} + `y` = `x`^{3} − 2`x` − 1,
the only curve with trivial torsion group (*i.e.*, the only torsion point on this
curve is the identity).

The data is as follows:

Video № | Ballet Move | Point | Curve equation | Order | Letter |
---|---|---|---|---|---|

1 | FOUETTE | (5, 5) | y^{2} + y = x^{3} − x^{2} − 10x − 20
| 5 | T |

2 | CHAINES | (5, 0) | y^{2} = x^{3} − x^{2} + 16x − 180
| 2 | H |

3 | TOUR JETE | (−1, 1) | y^{2} = x^{3} + x^{2} − x
| 6 | E |

4 | PIROUETTE | (−1, 2) | y^{2} + xy + y = x^{3} − x^{2} − 3x + 3
| 7 | T |

5 | WALTZ TURN | (0, 1) | y^{2} + y = x^{3} − 2x − 1
| 1 | W |

6 | PIQUE TURN | (7, 0) | y^{2} = x^{3} + x^{2} − 36x − 140
| 2 | I |

7 | PAS DE BOURREE(en tournant) | (0, −1) | y^{2} + y = x^{3} + x^{2} − x
| 3 | S |

8 | SOUTENU | (2, 4) | y^{2} = x^{3} + 4x
| 4 | T |

Those excellent dancers are Callie Norberg and Mae Chesney. Costumes by Rachel Petterson.