We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular
Let
For
Given some ring
The mass of the genus of Eisenstein lattices of rank
The mass was computed in [
To obtain the complete list of Eisenstein lattices of rank 16, we first constructed some lattices as orthogonal sums of Eisenstein lattices of rank 12 and 4 and from known 32dimensional even unimodular lattices. We also applied coding constructions from ternary and quaternary codes in the same spirit as described in [
There are exactly 83 isometry classes of
Since
Alternatively, one can classify these lattices directly using the neighbor method and a mass formula, which can be derived from the mass formula in [
The Eisenstein lattices of rank up to 16 are listed in Tables
The lattice of rank 4.
no. 





1 

155520  1 

The lattice of rank 8.
no. 





1  2 
48372940800  2  2 
The lattices of rank 12.
no. 





1  3 
22568879259648000  2  3 
2  4 
8463329722368  1  
3  6 
206391214080  1 

4  12 
101016305280  1  
5 

2690072985600  1 

The lattices of rank 16.
no. 





1  4 
14039648409841827840000  3  4 
2  4 
1316217038422671360  1  
3  6 
32097961613721600  1 

4  12 
15710055797145600  1  
5  4 
2742118830047232  1  
6  4 
40122452017152  1  
7 

418360150720512000  1 

8  10 
71409344532480  1  
9  8 
443823666757632  2 

10  4 
313456656384  
11  13 
11604018486528  
12  6 
825564856320  1  
13  6 
48977602560  
14  4 
15479341056  1  
15  7 
21427701120  
16  16 
1851353376768  3  
17  8 
8707129344  1  
18  4 
1451188224  
19  4 
9795520512  
20  4 
82556485632  1 

21 

1277045637120  
22  6 
302330880  2  
23  9 
1836660096  
24 

22448067840  
25  4 
107495424  1  
26  7 
52907904  
27  10 
408146688  1  
28  6 
22674816  
29  2 
134369280  1  
30  5 
8398080  
31  8 
423263232  2  
32  8 
7558272  4  
33  4 
4478976  
34  2 
7644119040  1 

35  2 
656916480  1  
36  7 
1530550080  
37  7 
2834352  
38  3 
113374080  
39  3 
2519424  
40  6 
1679616  1  
41  6 
629856  2  
42  2 
1710720  
43  5 
139968  
44 

3265920  
45 

2426112  
46  4 
161243136  2  
47  4 
68024448  1  
48  4 
4199040  2  
49  4 
1399680  1  
50  4 
314928  
51  4 
139968  1  
52  4 
69984  3  
53 

660290641920  
54 

1813985280  
55 

87091200 


56 

1990656  
57  3 
58320  
58  3 
15552  
59  2 
606528  
60  2 
186624  1  
61  2 
41472  1  
62  2 
25920  
63  2 
18144  2  
64  2 
18144  2  
65  2 
16200  4  
66 

2204496  
67 

108864  
68 

3888  
69 

2916  
70 

303216721920  2 

71 

15552000  5 

72 

9289728  3 

73 

1658880  1  
74 

387072  3  
75 

29376  2  
76 

10368  1  
77 

8064  2  
78 

5760  4  
79 

4608  2  
80 

2592  3 
A list of the Gram matrices of the lattices is given in [
We have the following.
The
Each of the lattices listed previously is isometric to its conjugate. Hence the associated Hermitian theta series are symmetric Hermitian modular forms (cf. [