# Short Element and Small Representative

## `ShortElement(I): AlgEtQIdl -> AlgEtQElt`

Given an ideal $I$ returns a non-zerodivisor in $I$ with small coefficients (in the LLL sense). This is achieved by randomly picking an element with small coefficients in a LLL-reduced basis (wrt the T2 norm as a ${{\Bbb Z}}$-lattice).

## `SmallRepresentative(I): AlgEtQIdl -> AlgEtQIdl, AlgEtQElt`

Given a fractional $R$-ideal $I$, it returns an isomorphic ideal $a \cdot I$, and the element $a$, such that $a \cdot I$ is a subset of $R$, and the cardinality of $R/aI$ is small. This is achieved by computing the `ShortElement` $a$ of $(R:I)$. Note that if $I$ is invertible $R/aI$ is isomorphic to $(R:I)/aR$.