Biased $$2 \times 2$$ periodic Aztec diamond and an elliptic curve

2 Elliptic Curve Properties

We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.

Covering the Aztec Diamond

There are 107 non-isomorphic coverings of the Aztec diamond of order 5 by the 25 one-sided tetrasticks. In [6] Donald Knuth describes how to attack certain types of puzzles by computer. One type of puzzle he tried to solve is the covering of patterns by tetrasticks. Tetrasticks — or more general polysticks — were introduced by Brian Barwell [1]: Pieces made from a fixed number of line segments ...

The number of genus 2 covers of an elliptic curve

The main aim of this paper is to determine the number cN,D of genus 2 covers of an elliptic curve E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈ Div(E). In the case that D is reduced, this formula is due to Dijkgraaf. The basic technique here for determining cN,D is to exploit the geometry of a certain compactification C = CE,N of the universal genus 2 curve over the Hurwitz space H...

Hurwitz spaces of genus 2 covers of an elliptic curve

Let E be an elliptic curve over a field K of characteristic 6= 2 and let N > 1 be an integer prime to char(K). The purpose of this paper is to study the family of genus 2 covers of E of fixed degree N , i.e. those covers f : C → E for which C/K is a curve of genus 2 and deg(f) = N . Since we can (without loss of generality) restrict our attention those covers that are normalized in the sense of...

Constructing Elliptic Curve Cryptosystems in Characteristic 2