Classifying pseudo-ovals, translation generalized quadrangles, and elation Laguerre planes of small order

We provide classification results for translation generalized quadrangles of order less than or equal to 64, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in PG ( 3 n - 1 , 2 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{PG}(3n-1,2)$$\end{document} , for n = 3 , 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=3,4$$\end{document} , and that of the pseudo-ovals in PG ( 3 n - 1 , q ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{PG}(3n-1,q)$$\end{document} , for n = 5 , 6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=5,6$$\end{document} , such that one of the associated projective planes is Desarguesian.