Translation Generalized Quadrangles of Order (s,s2), s Even, and Eggs

Characterizations of classical eggs and the classical generalized quadrangle Q(5,s), s even, are given. The egg O(n,2n,q)=O of PG(4n−1,q), q even, is classical if and only if either O is good at some element and contains at least one pseudo-conic, or O contains at least two intersecting pseudo-conics. As application three characterizations of the classical generalized quadrangle Q(5,s), s even, are given.