Quaternary Even Positive Definite Quadratic Forms of Discriminant 4p Original Research Article

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