The Lattice E6
An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, RWTH Aachen University
(nebe@math.rwth-aachen.de)
and
Neil J. A. Sloane
(njasloane@gmail.com)
Last modified Fri Jul 18 13:15:50 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
MINIMAL_NORM
KISSING_NUMBER
HERMITE_NUMBER
BASIS
GLUE_VECTORS
GRAM
DIVISORS
GROUP_ORDER
GROUP_NAME
PROPERTIES
NOTES
LAST_LINE
-
NAME
E6
-
DIMENSION
6
-
DET
3
-
MINIMAL_NORM
2
-
KISSING_NUMBER
72
-
HERMITE_NUMBER
.166536635531E+01
-
BASIS
6 8
.000000000000E+00 -.100000000000E+01 .100000000000E+01
.000000000000E+00 .000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00 -.100000000000E+01
.100000000000E+01 .000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00 .000000000000E+00
-.100000000000E+01 .100000000000E+01 .000000000000E+00
.000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00 .000000000000E+00
.000000000000E+00 -.100000000000E+01 .100000000000E+01
.000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00 -.100000000000E+01
.100000000000E+01 .000000000000E+00
.500000000000E+00 .500000000000E+00 .500000000000E+00
.500000000000E+00 -.500000000000E+00 -.500000000000E+00
-.500000000000E+00 -.500000000000E+00
-
GLUE_VECTORS
3
.000000000000E+00 .000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00 .000000000000E+00
.000000000000E+00 .000000000000E+00
.000000000000E+00 -.666666666667E+00 -.666666666667E+00
.333333333333E+00 .333333333333E+00 .333333333333E+00
.333333333333E+00 .000000000000E+00
.000000000000E+00 .666666666667E+00 .666666666667E+00
-.333333333333E+00 -.333333333333E+00 -.333333333333E+00
-.333333333333E+00 .000000000000E+00
-
GRAM
6 6
2 -1 0 0 0 0
-1 2 -1 0 0 0
0 -1 2 -1 0 -1
0 0 -1 2 -1 0
0 0 0 -1 2 0
0 0 -1 0 0 2
-
DIVISORS
1 1 1 1 1 3
-
GROUP_ORDER
103680
-
GROUP_NAME
C2 x W(E6)
-
PROPERTIES
INTEGRAL =1
-
NOTES
x in E8 with sum x(i) = 0 and x(1) + x(8) = 0
# maple program to construct the min vecs of E6 in 6-D coords
with(linalg): gc(0):
w:=-1/2 + I*sqrt(3)/2; w2:=expand(w^2): w6:=-w2: th:=w-w2:
f1:= proc(x) [ Re(x[1]), Re(x[2]), Re(x[3]),
Im(x[1]), Im(x[2]), Im(x[3]) ]: end:
# the 72 min vecs of E6
t[1]:=[th,0,0];
t[2]:=[0,th,0];
t[3]:=[0,0,th];
nov:=0;
for i1 from 1 to 3 do
for i from 1 to 6 do
nov:=nov+1:
mvE6[nov]:=f1( evalm (expand(w6^i) * t[i1] ) );
od; od;
for i1 from 1 to 2 do
for i from 1 to 3 do
for j from 1 to 3 do
for k from 1 to 3 do
nov:=nov+1:
mvE6[nov]:=f1(evalm(expand( (-1)^i1*[w^i,w^j,w^k] )));
od; od; od; od;
print(`minimal vectors of E6`);
for i from 1 to 72 do lprint(evalf(mvE6[i])); od;
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe