**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.

Some corrections to these tables have been found by Chul-hee Lee
Minor corrections to Nipps tables of quaternary quadratic forms .

For the corrected information compiled by Chul-hee Lee see
github tables.

**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**

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**