Exponent vector ($\mathbf{e}(r)=(e_1,\dots ,e_B)$)

Relative to an ordered factor base $(p_1,\dots ,p_B)$, a positive integer $r={\prod }_j{p}_j^{e_j}$ has exponent vector $(e_1,\dots ,e_B)$. Multiplication of such integers adds their exponent vectors. Reducing every coordinate modulo two gives the parity vector used to find congruences of squares. The ordering and any sign coordinate must be used consistently across all rows.