Primitive Integral Quadratic forms of fixed discriminant Assume all the quadratic forms below are integral. Use your favorite definition of discriminant of a quadratic form(a rational multiple of a matrix associated to the coefficients of the quadratic form). 
For binary quadratic forms, Gauss has shown that the equivalence class of primitive binary quadratic forms is finite and that there is a composition law which turns the set of equivalence classes into a group. 
By equivalence of integral n-ary quadratic forms, I mean a change of variables whose matrix is in SL(n,Z). That is, we are considering 'proper equivalence.' 


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*Is the set of SL(3,Z)-equivalence classes of primitive integral ternary quadratic forms finite? Is there a list for those of small discriminant? How about for quaternary forms or in general, n-ary forms?

*I was able to find a resource online for quaternary forms but I was not able to determine from the resource whether the author lists representatives from each equivalence class of primitive quaternary quadratic forms or is it for positive-definite quaternary quadratic forms? Does SL(n,Z)-equivalence preserve positive-definiteness? So, are the representatives in each class chosen such that they are reduced, regular, primitive and positive-definite?  
 A: Equivalence preserves positivity. If $H$ is the Hessian matrix of second partial derivatives of a form, the Hessian of any equivalent form is $P^T H P,$ where $P$ has integer entries and determinant $\pm 1.$ This determinant matters for binary forms, makes no difference in odd dimension, and is usually ignored (but not always, I helped W. K. Chan with something) for even dimension of at least four.
Nipp's tables are for positive forms. 
For positive ternary forms, THIS begins with $x^2 + y^2 + z^2$ which is alone in its genus. Then THIS begins with $x^2 + y^2 + z^2 + yz + zx + xy$ which is also alone in its genus.
I know of no online lists of indefinite forms; people ask me from time to time. When the number of variables is odd, there is no change in using equivalence or proper equivalence, because any form goes to itself by minus the identity matrix, which has negative determinant. 
Probably the best thing to start with on indefinite ternary forms is Dickson (1939) Modern Elementary Theory of Numbers.  He finds all forms for certain small discriminants. 
For indefinite ternary forms, we know that a form is alone in its genus (perhaps not its discriminant) if the discriminant is not divisible by any $p^3$ for odd prime $p$ or by a high power of $2.$ The example that illustrates the cutoff point is due to Siegel, 
$$ x^2 - 2 y^2 + 64 z^2 $$
has another form in its genus. Pages 168 and 253 in Cassels.
The best book for actually proving that a form integrally represents the suspected numbers is CASSELS, especially the tables for the Hilbert Norm Residue symbol on pages 43-44, with the lemmas on isotropy on pages 58-60.
Here is a jpeg from Dickson, it shows the 102 diagonal positive regular forms and the numbers that are not represented by it. 

