Quaternary even positive definite quadratic forms of discriminant 4p

Let p>13 be a prime congruent to 1 module 4. Let g be the genus of a quaternary even positive definite Z-lattice of discriminant 4p whose 2-adic localization has a proper 2-modular Jordan component. We show that the orthogonal group of any lattice from g is generated by -1 and the symmetries with respect to the roots of the lattice. The class number of g is computed. Furthermore, we show that the theta series of degree two coming from the classes in g with non-trivial automorphism groups are linearly independent. (C) 1999 Academic Press.