Say you have a 3x3x3 matrix filled with numbers, including in the very center. So you have 27 numbers in a special 3-dimension configuration. Since there are three axis for such a cube, there are three ways of dividing such a cube into three 3×3 matrices and Once you do that you can get a cubic form by computing
which gives you a cubic equation in three variables, or in other words a genus one curve.
Actually you get three different genus one curves, since you do it along any axis. Turns out there are crazy interesting relationships between those curves, as well as in the space of all 3x3x3 cubes.
Some work by MathBabe Cathy O'Neil made its way into the paper of colleagues and they are going to use this paper as background to understand the average size of Selmer groups of elliptic curves.
Their paper is over 70 pages long, so not for the faint of heart, but I don't have to peer review it and it's the weekend so ...
Preprint: Manjul Bhargava, Wei Ho, 'Coregular spaces and genus one curves', arXiv:1306.4424