Gordon Nipp's Tables of Quaternary Quadratic Forms

 Keywords: tables, reduced, regular, primitive, positive definite, quaternary, quadratic forms, four-dimensional lattices, automorphism group, mass, genus, genera, Hasse symbol, Jordan splitting

 These tables were computed by Gordon L. Nipp (gnipp@calstatela.edu), of the Department of Mathematics, California State University, Los Angeles, CA 90032, USA, and were published in his book: Gordon L. Nipp, "Quaternary Quadratic Forms: Computer Generated Tables", Springer-Verlag, New York, 1991; ISBN 0-387-97601-9. They are included here with his permission.

Contents of these files

About these tables
About this Catalogue of Lattices
The format of these tables
Discriminants 4 to 457
Discriminants 458 to 641
Discriminants 642 to 777
Discriminants 778 to 893
Discriminants 894 to 992
Discriminants 993 to 1080
Discriminants 1081 to 1161
Discriminants 1162 to 1236
Discriminants 1237 to 1308
Discriminants 1309 to 1373
Discriminants 1374 to 1433
Discriminants 1434 to 1492
Discriminants 1493 to 1549
Discriminants 1550 to 1604
Discriminants 1605 to 1656
Discriminants 1657 to 1705
Discriminants 1706 to 1732
Genera for Discriminants 4 to 1080
Genera for Discriminants 1081 to 1732

Format of these files

These are tables of reduced regular primitive positive-definite quaternary quadratic forms over the rational integers.

They were computed by Gordon L. Nipp (see above) and are included here with his permission.

The discriminant d of a quaternary quadratic form

      f = f11 x1^2 + f22 x2^2 + f33 x3^2 + f44 x4^2
            + f12 x1 x2 + f13 x1 x3 + f23 x2 x3 + f14 x1 x4 + f24 x2 x4 + f34 x3 x4

where the coefficients f11, f12, ..., f44 are integers, is defined to be the determinant of the associated matrix F =

  [ 2f11   f12   f13   f14 ]
  [ f12   2f22   f23   f24 ]
  [ f13   f23   2f33   f34 ]
  [ f14   f24   f34   2f44 ]

Then d is an integer congruent to 0 or 1 mod 4.

In the discriminant tables there is one line for each form, containing 17 entries:

d   g   f11   f22   f33   f44   f12   f13   f23   f14   f24   f34   H   N   G   m1   m2  

where
      d = discriminant,
      g = number of genus to which f belongs,
      f11 ... f34 are the coefficients,
      H = Hasse symbol at all primes p dividing 2d, in increasing order of p (using 10 characters),
      N = level of   f = smallest N such that N F^(-1) has integer entries and even diagonal entries,
      G = number of automorphisms of   f,
      m1/m2 = the total mass of this genus

As a check, the sum of 1/G over all forms with the same first two entries (i.e. the same genus) should equal m1/m2

The two genera files give further information about each genus. For each discriminant d and each genus, there is a line for each prime p dividing 2d, giving the p-adic density, and a p-adic Jordan splitting for the first form listed in the genus.

See Nipp's book mentioned above for further information about these tables.

About these tables | About this Catalogue of Lattices