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Ideal lattices in the plane coming from real quadratic number fields havebeen investigated by several authors in the recent years. In particular, it hasbeen proved that every such ideal has a basis that can be twisted by the actionof the diagonal group into a Minkowski reduced basis for a well-roundedlattice. We explicitly study such twists on the canonical bases of ideals,which are especially important in arithmetic theory of quadratic number fieldsand binary quadratic forms. Specifically, we prove that every fixed realquadratic field has only finitely many ideals whose canonical basis can betwisted into a well-rounded or a stable lattice in the plane. We demonstratesome explicit examples of such twists. We also briefly discuss the relationbetween stable and well-rounded twists of arbitrary ideal bases.

Canonical basis twists of ideal lattices from real quadratic number fields