How to create a lattice (a matrix) in Sage

Type the rows of the matrix one by one.

Take a determinant

Type the rows of the matrix one by one.

Extract a column vector from a matrix

The integer in the notation below is the number of the column. 0 = first column, 1 = second column, etc.

Create, add and multiply vectors

Take a dot product of vectors

Verify the shortest vector in a lattice

Here’s a little tool to graph a 2D lattice (initialization cell)

Here's how to use the graphing tool

The first variable is the matrix, column vectors = lattice.
The second variable "box" tells you what width of box to show.
The third variable "search" expands the search for lattices point in the box.
If you aren't seeing the lattice fill the box, make the box smaller or the search bigger.