Motivated by the ring of integers of cyclic number fields of prime degree, we introduce the notion of Lagrangian lattices. Furthermore, given an arbitrary non-trivial lattice L we construct a family of full-rank sub-lattices {Lα} of L such that whenever L is Lagrangian it can be easily checked whether or not Lα has a basis of minimal vectors. In this case, a basis of minimal vectors of Lα is given.

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We study ideal lattices in R 2 coming from real quadratic fields, and give an explicit method for computing all well-rounded twists of any such ideal lattice. We apply this to ideal lattices coming from Markoff numbers to construct infinite families of non-equivalent planar lattices with good sphere-packing radius and good minimum product distance. We also provide a complete classification of all real quadratic fields such that the orthogonal lattice is the only well-rounded twist of the lattice corresponding to the ring of integers.

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