The Square Curve

The square curve is an elliptic curve I used to check elliptic curve arithmetic while writing the socelliptic library. It is a Mordell curve with equation y²=x³+1 over F₃₄₃. All real elliptic curves have one or two branches, so at the time I thought that all elliptic curves over finite fields have one or two branches, too. Imagine my astonishment when this curve turned out to have 18 branches! (I'm using the term "branches" like branches of a hyperbola, not branches of the square root function. By the latter definition, all elliptic curves have two branches.)

The graph at the left, and magnified below, shows the curve and its twist in the real plane, with the rational points as bigger circles. The 17 points on the straight (black) curve and the point at infinity correspond to the 18 points on the main branch of the square curve. The 13 points on the twisted (blue) curve are just some of the points on the main branch of the twist. The twist is hard to imagine, as it has two branches, the other passing through the other cube roots of 1 (which are 2 and 4 in F₃₄₃), yet it is symmetric (as is the straight curve) by rotating it by 120° in the complex plane (1->2->4->).