# On one class of Euclidean lattices

Let $\Lambda\subset \mathbb Z^3$ be 3D lattice with a basis $$a_1=\left(\begin{smallmatrix} a_{11} \\ a_{21}\\ a_{31} \end{smallmatrix}\right),a_2=\left(\begin{smallmatrix} a_{12} \\ a_{22}\\ a_{32} \end{smallmatrix}\right),a_3=\left(\begin{smallmatrix} a_{13} \\ a_{23}\\ a_{33} \end{smallmatrix}\right).$$ Is there a special name for lattices such that minors $A_{ij}$ of matrix $A=(a_{ij})_{i,j=1}^3$ satisfy an additional restriction $$GCD(A_{i1},A_{i2},A_{i3})=1\qquad(i=1,2,3)?$$

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Is the lattice integral? Does the bracket indicate a GCD? – Wilberd van der Kallen Jan 9 '14 at 8:18
@Wilberd van der Kallen Sorry. Yes. Yes. – Alexey Ustinov Jan 9 '14 at 9:44
I suspect you are asking about the "reciprocal" form, which is a special case of Watson's transformations. Please see pdf Watson_Transformations_1 at my zakuski.math.utsa.edu/~kap/forms.html and let me know. I can probably help with this, but I don't know what you are after yet. – Will Jagy Jan 9 '14 at 20:29
@Will Jagy Yes it's some kind of "reciprocal" condition. But I don't even know the name for lattices with $(a_{11},a_{12},a_{13})=1$,... – Alexey Ustinov Jan 10 '14 at 10:30