[was@modular lattices]$ [was@modular lattices]$ Magma V2.8-1      Thu Aug  9 2001 15:24:36 on modular  [Seed = 1]
Linked at:        Fri Aug 03 2001 22:12:01
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
>  f := x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 \
           + x^7 - x^6 - x^3 + x^2 - x + 1;
> > 
> 
> RR<y> := PolynomialRing(RealField());
> rr := Roots(RR!f);
> rr;
[ <1.18836814750822358814296095862959359470470455961, 1>, <0.841490073675237018423791922905132477394891071739, 1>, <0.910643668480331802841280039503848289906854066953 + 0.413192581076528461640690252083950737182707437166*i, 1>, <0.910643668480331802841280039503848289906854066957 - 0.413192581076528461640690252083950737182707437154*i, 1>, <0.613776083585744894315249515043812887060178799144 - 0.789480157583548579027964312518863454650723636936*i, 1>, <0.613776083585744894315249515043812887060178799144 + 0.789480157583548579027964312518863454650723636936*i, 1>, <-0.779446249023135419112103356900558753082764941039 + 0.626469109281347062101002045542795190706651111300*i, 1>, <-0.779446249023135419112103356900558753082764941031 - 0.626469109281347062101002045542795190706651111299*i, 1>, <-0.450404336901007751603533702096947845452249737458 + 0.892824693487340496174538044304238781491584421392*i, 1>, <-0.450404336901007751603533702096947845452249737454 - 0.892824693487340496174538044304238781491584421391*i, 1>, <-0.997624669509481824916568433820703872690953759838 + 0.0688840967575039402331779007336972122263791519406*i, 1>, <-0.997624669509481824916568433820703872690953759981 - 0.0688840967575039402331779007336972122263791519372*i, 1>, <0.353886995687930931828260975279761521301582440665 + 0.935288187823929675262425108562184309174706113289*i, 1>, <0.353886995687930931828260975279761521301582440534 - 0.935288187823929675262425108562184309174706113034*i, 1>, <0.103259607299440299200534769711329852138868337415 + 0.994654439240264676796857447734137643840433394385*i, 1>, <0.103259607299440299200534769711329852138868337589 - 0.994654439240264676796857447734137643840433394830*i, 1>, <-0.269020210211553235836496247487905115231313021478 + 0.963134531878974687244819195078347609855924982770*i, 1>, <-0.269020210211553235836496247487905115231313021517 - 0.963134531878974687244819195078347609855924982590*i, 1> ]
> rr[1];
<1.18836814750822358814296095862959359470470455961, 1>
> rr[2];
<0.841490073675237018423791922905132477394891071739, 1>
> rr[3];
<0.910643668480331802841280039503848289906854066953 + 0.413192581076528461640690252083950737182707437166*i, 1>
> rr[4];
<0.910643668480331802841280039503848289906854066957 - 0.413192581076528461640690252083950737182707437154*i, 1>
> Parent(rr[2][1]);
> Parent(rr[2][1]);
Real Field
> [r[1] : r in rr | Parent(r[1]) eq RealField()];
[ 1.18836814750822358814296095862959359470470455961, 0.841490073675237018423791922905132477394891071739 ]
> a := $1;
> a;
[ 1.18836814750822358814296095862959359470470455961, 0.841490073675237018423791922905132477394891071739 ]
> #a;
2
> a[1]+1/a[1];
2.02985822118346060656675288153472607209959563154
> a[2]+1/a[2];
2.02985822118346060656675288153472607209959563161
> a[1];
1.18836814750822358814296095862959359470470455961
> 1/a[1];
0.841490073675237018423791922905132477394891071922
> g := x^9 - 3*x^8 - 23*x^7 + 35*x^6 + 81*x^5 - 116*x^4 - 67*x^3 + 104*x^2 + 8*x - 19;
> Roots(RR!g);
[ <-0.468111906454000689322313234631336807368973901776, 1>, <0.558484342299467301848260817414060396183155100382, 1>, <1.2313073740889668988677367163865053658, 1>, <1.3706568287964283199650620889897443169, 1>, <1.1474930126118010257418142242557119149, 1>, <5.6479185634867740095959132409297992529, 1>, <-3.6958681103558892478674250772828809120, 1>, <-1.7843250602092410500834767489211165859, 1>, <-1.00755504426430656874557202714048694160132114234, 1> ]
> Evaluate(RR!g,a[1]+1/a[1]);
-409.723792971540941413127157806997691369740188522
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 




>  f := x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 
          + x^7 - x^6 - x^3 + x^2 - x + 1;
> K<a> := NumberField(f);
> g := MinimalPolynomial(a+1/a);
> g;
x^9 - x^8 - 8*x^7 + 7*x^6 + 20*x^5 - 14*x^4 - 17*x^3 + 8*x^2 + 4*x - 1
> L<b> := NumberField(g);
> L;
Number Field with defining polynomial x^9 - x^8 - 8*x^7 + 7*x^6 + 20*x^5 - 14*x^4 - 17*x^3 + 8*x^2 + 4*x - 1 over the Rational Field
> time h := ClassNumber(MaximalOrder(L));
Time: 0.900
> h;
1
> O := MaximalOrder(L);
> time U,phi := UnitGroup(O);  // takes < 10 seconds
Time: 2.800
> U;   // as expected (r-1)
Abelian Group isomorphic to Z/2 + Z (8 copies)
Defined on 9 generators
Relations:
2*U.1 = 0
> gprime := Derivative(g); gprime;
9*x^8 - 8*x^7 - 56*x^6 + 42*x^5 + 100*x^4 - 56*x^3 - 51*x^2 + 16*x + 4
> alpha := 1/Evaluate(gprime,b);
> RR<y> := PolynomialRing(RealField());
> #[r : r in Roots(RR!MinimalPolynomial(alpha)) | r[1] gt 0];
5
> R := [r[1] : r in Roots(RR!g)];  // these correspond to the real embeddings.
> function EmbeddingSigns(z)
    h := RR!Eltseq(L!z);
    return [Sign(Evaluate(h, R[i])) : i in [1..#R]];
  end function;
function> function> function> > 
> 
> EmbeddingSigns(alpha);
[ -1, 1, -1, 1, -1, 1, 1, -1, 1 ]
> for i in [1..9] do EmbeddingSigns(phi(U.i)); end for;
[ -1, -1, -1, -1, -1, -1, -1, -1, -1 ]
[ 1, 1, -1, 1, 1, 1, -1, -1, -1 ]
[ 1, 1, -1, -1, -1, -1, 1, 1, -1 ]
[ 1, 1, 1, -1, -1, 1, 1, -1, -1 ]
[ 1, 1, 1, -1, -1, -1, 1, 1, 1 ]
[ 1, -1, -1, -1, 1, -1, 1, -1, 1 ]
[ 1, -1, -1, -1, 1, 1, 1, -1, 1 ]
[ 1, 1, 1, 1, -1, -1, -1, -1, -1 ]
[ 1, -1, -1, 1, 1, -1, -1, 1, -1 ]
>  A := MatrixAlgebra(GF(2),9)!0;
> for r in [1..9] do for c in [1..9] do 
> for|for>      if EmbeddingSigns(phi(U.r))[c] eq -1 then A[r,c] := 1; end if;
  end for; end for;
for|for> > 
> A;
[1 1 1 1 1 1 1 1 1]
[0 0 1 0 0 0 1 1 1]
[0 0 1 1 1 1 0 0 1]
[0 0 0 1 1 0 0 1 1]
[0 0 0 1 1 1 0 0 0]
[0 1 1 1 0 1 0 1 0]
[0 1 1 1 0 0 0 1 0]
[0 0 0 0 1 1 1 1 1]
[0 1 1 0 0 1 1 0 1]
> V := RowSpace(A);
> V;
Vector space of degree 9, dimension 8 over GF(2)
Echelonized basis:
(1 0 0 0 0 0 0 0 1)
(0 1 0 0 0 0 0 0 1)
(0 0 1 0 0 0 0 0 1)
(0 0 0 1 0 0 0 0 1)
(0 0 0 0 1 0 0 0 1)
(0 0 0 0 0 1 0 0 0)
(0 0 0 0 0 0 1 0 1)
(0 0 0 0 0 0 0 1 1)
> W := VectorSpace(GF(2),9);
> // [ -1, 1, -1, 1, -1, 1, 1, -1, 1 ]
> v := W![1,0,1,0,1,0,0,1,0];
> > v in V;
true
> v;
(1 0 1 0 1 0 0 1 0)
> Eltseq(V!v);
[ 1, 0, 1, 0, 1, 0, 0, 1, 0 ]
> ?Solve;
MAGMA_HTML_DIR is not set
> Solution(A,v);
(1 1 0 0 0 1 0 0 0)
> alpha;
1/53995179961*(8462152596*b^8 - 6852528273*b^7 - 64893174399*b^6 + 44327911581*b^5 + 148290056856*b^4 - 74228933970*b^3 - 99225969383*b^2 + 23785291597*b + 10873550202)
> d := alpha*phi(U.1+U.2+U.6);
> EmbeddingSigns(d);
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
> d;
1/53995179961*(-3426147347*b^8 + 817946757*b^7 + 29470572047*b^6 - 3984702109*b^5 - 80455461231*b^4 + 2178527185*b^3 + 70870946633*b^2 + 3409710051*b - 2902988849)
> Norm(d);
1/53995179961
> Norm(alpha);
1/53995179961
> // d is the totally positive generator for the inverse different!
> Kernel(A);
Vector space of degree 9, dimension 1 over GF(2)
Echelonized basis:
(0 0 0 0 1 1 0 1 1)
> // ... and, that's a nonsquare unit that is totally positive...
> 
> O;
Maximal Equation Order with defining polynomial x^9 - x^8 - 8*x^7 + 7*x^6 + 20*x^5 - 14*x^4 - 17*x^3 + 8*x^2 + 4*x - 1 over Z
> Trace(d*O.1*O.2);
0
> Trace(d*O.1*O.3);
1
> function bin(a,b) return Trace(d*a*b); end function;
> bin(O.1,O.3);
1
> Type($1);
FldRatElt
> B := MatrixAlgebra(Rationals(),9)!0;
> for r in [1..9] do for c in [1..9] do B[r,c] := bin(O.r,O.c); end for; end for;
> B;
[   1    0    1    0    2    0    6    0   21]
[   0    1    0    2    0    6    0   21    0]
[   1    0    2    0    6    0   21    0   78]
[   0    2    0    6    0   21    0   78    0]
[   2    0    6    0   21    0   78    0  298]
[   0    6    0   21    0   78    0  298    0]
[   6    0   21    0   78    0  298    0 1157]
[   0   21    0   78    0  298    0 1157    1]
[  21    0   78    0  298    0 1157    1 4539]
> Determinant(B);
1
> // B is the Gram matrix that defines our lattice.
> L := LatticeWithGram(B);
> L;
Standard Lattice of rank 9 and degree 9
Inner Product Matrix:
[   1    0    1    0    2    0    6    0   21]
[   0    1    0    2    0    6    0   21    0]
[   1    0    2    0    6    0   21    0   78]
[   0    2    0    6    0   21    0   78    0]
[   2    0    6    0   21    0   78    0  298]
[   0    6    0   21    0   78    0  298    0]
[   6    0   21    0   78    0  298    0 1157]
[   0   21    0   78    0  298    0 1157    1]
[  21    0   78    0  298    0 1157    1 4539]
> time short := ShortVectors(L,1);
Time: 0.000
> short;
[
<( 1  0 -1  0  0  0  0  0  0), 1>,
<(1 0 0 0 0 0 0 0 0), 1>,
<( 2  0 -4  0  1  0  0  0  0), 1>,
<( 3  0 -9  0  6  0 -1  0  0), 1>,
<(  4   0 -17   0  20   0  -8   0   1), 1>,
<(  4   3 -17  -9  20   6  -8  -1   1), 1>,
<(  4   5 -17 -13  20   7  -8  -1   1), 1>,
<(  4   7 -17 -14  20   7  -8  -1   1), 1>,
<(0 1 0 0 0 0 0 0 0), 1>
]
> MinimalPolynomial(d);
x^9 - x^8 + 21031903322/53995179961*x^7 - 4130791902/53995179961*x^6 + 447388898/53995179961*x^5 - 27890118/53995179961*x^4 + 1015898/53995179961*x^3 - 21138/53995179961*x^2 + 230/53995179961*x - 1/53995179961
> 

> restore "august9.session";
Restoring Magma state from "august9.session"
> d;
1/53995179961*(-3426147347*b^8 + 817946757*b^7 + 29470572047*b^6 - 3984702109*b^5 - 80455461231*b^4 + 2178527185*b^3 + 70870946633*b^2 + 3409710051*b - 2902988849)
> EmbeddingSigns(b+2);
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
> d2 := (b+2)*d;
> function bin2(a,b) return Trace(d2*a*b); end function;
> B2 := MatrixAlgebra(Rationals(),9)!0;
> for r in [1..9] do for c in [1..9] do B2[r,c] := bin2(O.r,O.c); end for; end for;
> B2;
[   2    1    2    2    4    6   12   21   42]
[   1    2    2    4    6   12   21   42   78]
[   2    2    4    6   12   21   42   78  156]
[   2    4    6   12   21   42   78  156  298]
[   4    6   12   21   42   78  156  298  596]
[   6   12   21   42   78  156  298  596 1157]
[  12   21   42   78  156  298  596 1157 2315]
[  21   42   78  156  298  596 1157 2315 4541]
[  42   78  156  298  596 1157 2315 4541 9095]
> 
> L2 :=  LatticeWithGram(B2);
> L2;
Standard Lattice of rank 9 and degree 9
Inner Product Matrix:
[   2    1    2    2    4    6   12   21   42]
[   1    2    2    4    6   12   21   42   78]
[   2    2    4    6   12   21   42   78  156]
[   2    4    6   12   21   42   78  156  298]
[   4    6   12   21   42   78  156  298  596]
[   6   12   21   42   78  156  298  596 1157]
[  12   21   42   78  156  298  596 1157 2315]
[  21   42   78  156  298  596 1157 2315 4541]
[  42   78  156  298  596 1157 2315 4541 9095]
>  time short2 := ShortVectors(L2,1);
Time: 0.000
> #short2;
1
> short2;
[
<( 2 -1 -8  4  8 -4 -2  1  0), 1>
]
> Determinant(L2);
1
> [was@modular lattices]$ [was@modular lattices]$ Magma V2.8-1      Thu Aug  9 2001 16:12:36 on modular  [Seed = 1]
Linked at:        Fri Aug 03 2001 22:12:01
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> restore "august9.session";
Restoring Magma state from "august9.session"
> time s2 := ShortVectors(L2,2);
Time: 0.000
> #s2;
121
> time #ShortVectors(L2,3);
361
Time: 0.000
> time #ShortVectors(L2,4);
1442
Time: 0.010
> time #ShortVectors(L2,5);
3602
Time: 0.040
> time #ShortVectors(L,2);
121
Time: 0.000
> time L := LatticeWithGram(B);
Time: 0.000
> time #ShortVectors(L,2);
81
Time: 0.000
> time ShortVectors(L,2);
[
<( 1  0 -1  0  0  0  0  0  0), 1>,
<(1 0 0 0 0 0 0 0 0), 1>,
<( 2  0 -4  0  1  0  0  0  0), 1>,
<( 3  0 -9  0  6  0 -1  0  0), 1>,
<(  4   0 -17   0  20   0  -8   0   1), 1>,
<(  4   3 -17  -9  20   6  -8  -1   1), 1>,
<(  4   5 -17 -13  20   7  -8  -1   1), 1>,
<(  4   7 -17 -14  20   7  -8  -1   1), 1>,
<(0 1 0 0 0 0 0 0 0), 1>,
<( 1 -1 -1  0  0  0  0  0  0), 2>,
<( 1 -1  0  0  0  0  0  0  0), 2>,
<( 1  0 -8  0 14  0 -7  0  1), 2>,
<( 1  0 -5  0  5  0 -1  0  0), 2>,
<( 1  0 -4  0  1  0  0  0  0), 2>,
<( 1  0 -3  0  1  0  0  0  0), 2>,
<( 1  1 -1  0  0  0  0  0  0), 2>,
<(1 1 0 0 0 0 0 0 0), 2>,
<( 1  3 -8 -9 14  6 -7 -1  1), 2>,
<(  1   5  -8 -13  14   7  -7  -1   1), 2>,
<(  1   7  -8 -14  14   7  -7  -1   1), 2>,
<( 2 -1 -4  0  1  0  0  0  0), 2>,
<(  2   0 -13   0  19   0  -8   0   1), 2>,
<( 2  0 -9  0  6  0 -1  0  0), 2>,
<( 2  0 -8  0  6  0 -1  0  0), 2>,
<( 2  0 -1  0  0  0  0  0  0), 2>,
<( 2  1 -4  0  1  0  0  0  0), 2>,
<(  2   3 -13  -9  19   6  -8  -1   1), 2>,
<(  2   5 -13 -13  19   7  -8  -1   1), 2>,
<(  2   7 -13 -14  19   7  -8  -1   1), 2>,
<( 3 -1 -9  0  6  0 -1  0  0), 2>,
<(  3   0 -17   0  20   0  -8   0   1), 2>,
<(  3   0 -16   0  20   0  -8   0   1), 2>,
<( 3  0 -5  0  1  0  0  0  0), 2>,
<( 3  0 -4  0  1  0  0  0  0), 2>,
<( 3  1 -9  0  6  0 -1  0  0), 2>,
<(  3   3 -17  -9  20   6  -8  -1   1), 2>,
<(  3   3 -16  -9  20   6  -8  -1   1), 2>,
<(  3   5 -17 -13  20   7  -8  -1   1), 2>,
<(  3   5 -16 -13  20   7  -8  -1   1), 2>,
<(  3   7 -17 -14  20   7  -8  -1   1), 2>,
<(  3   7 -16 -14  20   7  -8  -1   1), 2>,
<(  4  -1 -17   0  20   0  -8   0   1), 2>,
<(  4   0 -10   0   6   0  -1   0   0), 2>,
<( 4  0 -9  0  6  0 -1  0  0), 2>,
<(  4   1 -17   0  20   0  -8   0   1), 2>,
<(  4   2 -17  -9  20   6  -8  -1   1), 2>,
<(  4   4 -17 -13  20   7  -8  -1   1), 2>,
<(  4   4 -17  -9  20   6  -8  -1   1), 2>,
<(  4   6 -17 -14  20   7  -8  -1   1), 2>,
<(  4   6 -17 -13  20   7  -8  -1   1), 2>,
<(  4   8 -17 -14  20   7  -8  -1   1), 2>,
<(  5   0 -18   0  20   0  -8   0   1), 2>,
<(  5   0 -17   0  20   0  -8   0   1), 2>,
<(  5   0 -13   0   7   0  -1   0   0), 2>,
<(  5   3 -18  -9  20   6  -8  -1   1), 2>,
<(  5   3 -17  -9  20   6  -8  -1   1), 2>,
<(  5   5 -18 -13  20   7  -8  -1   1), 2>,
<(  5   5 -17 -13  20   7  -8  -1   1), 2>,
<(  5   7 -18 -14  20   7  -8  -1   1), 2>,
<(  5   7 -17 -14  20   7  -8  -1   1), 2>,
<(  6   0 -21   0  21   0  -8   0   1), 2>,
<(  6   3 -21  -9  21   6  -8  -1   1), 2>,
<(  6   5 -21 -13  21   7  -8  -1   1), 2>,
<(  6   7 -21 -14  21   7  -8  -1   1), 2>,
<(  7   0 -26   0  26   0  -9   0   1), 2>,
<(  7   3 -26  -9  26   6  -9  -1   1), 2>,
<(  7   5 -26 -13  26   7  -9  -1   1), 2>,
<(  7   7 -26 -14  26   7  -9  -1   1), 2>,
<(  8   3 -34  -9  40   6 -16  -1   2), 2>,
<(  8   5 -34 -13  40   7 -16  -1   2), 2>,
<(  8   7 -34 -14  40   7 -16  -1   2), 2>,
<(  8   8 -34 -22  40  13 -16  -2   2), 2>,
<(  8  10 -34 -23  40  13 -16  -2   2), 2>,
<(  8  12 -34 -27  40  14 -16  -2   2), 2>,
<( 0  2  0 -4  0  1  0  0  0), 2>,
<( 0  2  0 -1  0  0  0  0  0), 2>,
<( 0  3  0 -9  0  6  0 -1  0), 2>,
<( 0  4  0 -5  0  1  0  0  0), 2>,
<(  0   5   0 -13   0   7   0  -1   0), 2>,
<(  0   7   0 -14   0   7   0  -1   0), 2>,
<(0 0 1 0 0 0 0 0 0), 2>
]
Time: 0.010
> save "august9.session";
Saving Magma state to "august9.session"
> quit;

Total time: 2.790 seconds
[was@modular lattices]$ 




[was@modular lattices]$ [was@modular lattices]$ Magma V2.8-1      Thu Aug  9 2001 17:13:30 on modular  [Seed = 1]
Linked at:        Fri Aug 03 2001 22:12:01
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> restore "august9.session";
Restoring Magma state from "august9.session"
> L2;
Standard Lattice of rank 9 and degree 9
Inner Product Matrix:
[   2    1    2    2    4    6   12   21   42]
[   1    2    2    4    6   12   21   42   78]
[   2    2    4    6   12   21   42   78  156]
[   2    4    6   12   21   42   78  156  298]
[   4    6   12   21   42   78  156  298  596]
[   6   12   21   42   78  156  298  596 1157]
[  12   21   42   78  156  298  596 1157 2315]
[  21   42   78  156  298  596 1157 2315 4541]
[  42   78  156  298  596 1157 2315 4541 9095]
> time ShortVectors(L,2);
[
<( 1  0 -1  0  0  0  0  0  0), 1>,
<(1 0 0 0 0 0 0 0 0), 1>,
<( 2  0 -4  0  1  0  0  0  0), 1>,
<( 3  0 -9  0  6  0 -1  0  0), 1>,
<(  4   0 -17   0  20   0  -8   0   1), 1>,
<(  4   3 -17  -9  20   6  -8  -1   1), 1>,
<(  4   5 -17 -13  20   7  -8  -1   1), 1>,
<(  4   7 -17 -14  20   7  -8  -1   1), 1>,
<(0 1 0 0 0 0 0 0 0), 1>,
<( 1 -1 -1  0  0  0  0  0  0), 2>,
<( 1 -1  0  0  0  0  0  0  0), 2>,
<( 1  0 -8  0 14  0 -7  0  1), 2>,
<( 1  0 -5  0  5  0 -1  0  0), 2>,
<( 1  0 -4  0  1  0  0  0  0), 2>,
<( 1  0 -3  0  1  0  0  0  0), 2>,
<( 1  1 -1  0  0  0  0  0  0), 2>,
<(1 1 0 0 0 0 0 0 0), 2>,
<( 1  3 -8 -9 14  6 -7 -1  1), 2>,
<(  1   5  -8 -13  14   7  -7  -1   1), 2>,
<(  1   7  -8 -14  14   7  -7  -1   1), 2>,
<( 2 -1 -4  0  1  0  0  0  0), 2>,
<(  2   0 -13   0  19   0  -8   0   1), 2>,
<( 2  0 -9  0  6  0 -1  0  0), 2>,
<( 2  0 -8  0  6  0 -1  0  0), 2>,
<( 2  0 -1  0  0  0  0  0  0), 2>,
<( 2  1 -4  0  1  0  0  0  0), 2>,
<(  2   3 -13  -9  19   6  -8  -1   1), 2>,
<(  2   5 -13 -13  19   7  -8  -1   1), 2>,
<(  2   7 -13 -14  19   7  -8  -1   1), 2>,
<( 3 -1 -9  0  6  0 -1  0  0), 2>,
<(  3   0 -17   0  20   0  -8   0   1), 2>,
<(  3   0 -16   0  20   0  -8   0   1), 2>,
<( 3  0 -5  0  1  0  0  0  0), 2>,
<( 3  0 -4  0  1  0  0  0  0), 2>,
<( 3  1 -9  0  6  0 -1  0  0), 2>,
<(  3   3 -17  -9  20   6  -8  -1   1), 2>,
<(  3   3 -16  -9  20   6  -8  -1   1), 2>,
<(  3   5 -17 -13  20   7  -8  -1   1), 2>,
<(  3   5 -16 -13  20   7  -8  -1   1), 2>,
<(  3   7 -17 -14  20   7  -8  -1   1), 2>,
<(  3   7 -16 -14  20   7  -8  -1   1), 2>,
<(  4  -1 -17   0  20   0  -8   0   1), 2>,
<(  4   0 -10   0   6   0  -1   0   0), 2>,
<( 4  0 -9  0  6  0 -1  0  0), 2>,
<(  4   1 -17   0  20   0  -8   0   1), 2>,
<(  4   2 -17  -9  20   6  -8  -1   1), 2>,
<(  4   4 -17 -13  20   7  -8  -1   1), 2>,
<(  4   4 -17  -9  20   6  -8  -1   1), 2>,
<(  4   6 -17 -14  20   7  -8  -1   1), 2>,
<(  4   6 -17 -13  20   7  -8  -1   1), 2>,
<(  4   8 -17 -14  20   7  -8  -1   1), 2>,
<(  5   0 -18   0  20   0  -8   0   1), 2>,
<(  5   0 -17   0  20   0  -8   0   1), 2>,
<(  5   0 -13   0   7   0  -1   0   0), 2>,
<(  5   3 -18  -9  20   6  -8  -1   1), 2>,
<(  5   3 -17  -9  20   6  -8  -1   1), 2>,
<(  5   5 -18 -13  20   7  -8  -1   1), 2>,
<(  5   5 -17 -13  20   7  -8  -1   1), 2>,
<(  5   7 -18 -14  20   7  -8  -1   1), 2>,
<(  5   7 -17 -14  20   7  -8  -1   1), 2>,
<(  6   0 -21   0  21   0  -8   0   1), 2>,
<(  6   3 -21  -9  21   6  -8  -1   1), 2>,
<(  6   5 -21 -13  21   7  -8  -1   1), 2>,
<(  6   7 -21 -14  21   7  -8  -1   1), 2>,
<(  7   0 -26   0  26   0  -9   0   1), 2>,
<(  7   3 -26  -9  26   6  -9  -1   1), 2>,
<(  7   5 -26 -13  26   7  -9  -1   1), 2>,
<(  7   7 -26 -14  26   7  -9  -1   1), 2>,
<(  8   3 -34  -9  40   6 -16  -1   2), 2>,
<(  8   5 -34 -13  40   7 -16  -1   2), 2>,
<(  8   7 -34 -14  40   7 -16  -1   2), 2>,
<(  8   8 -34 -22  40  13 -16  -2   2), 2>,
<(  8  10 -34 -23  40  13 -16  -2   2), 2>,
<(  8  12 -34 -27  40  14 -16  -2   2), 2>,
<( 0  2  0 -4  0  1  0  0  0), 2>,
<( 0  2  0 -1  0  0  0  0  0), 2>,
<( 0  3  0 -9  0  6  0 -1  0), 2>,
<( 0  4  0 -5  0  1  0  0  0), 2>,
<(  0   5   0 -13   0   7   0  -1   0), 2>,
<(  0   7   0 -14   0   7   0  -1   0), 2>,
<(0 0 1 0 0 0 0 0 0), 2>
]
Time: 0.010
> s3 := ShortVectors(L,3);
> #s3;
417
> 2*#ShortVectors(L,3) - 2*#ShortVectors(L,2);
672
> s3 := ShortVectors(L,3);
> s3[1];
<( 1  0 -1  0  0  0  0  0  0), 1>
> B2;
[   2    1    2    2    4    6   12   21   42]
[   1    2    2    4    6   12   21   42   78]
[   2    2    4    6   12   21   42   78  156]
[   2    4    6   12   21   42   78  156  298]
[   4    6   12   21   42   78  156  298  596]
[   6   12   21   42   78  156  298  596 1157]
[  12   21   42   78  156  298  596 1157 2315]
[  21   42   78  156  298  596 1157 2315 4541]
[  42   78  156  298  596 1157 2315 4541 9095]
> Q := QuadraticForm(L2);
> Q;
2*x1^2 + 2*x1*x2 + 4*x1*x3 + 4*x1*x4 + 8*x1*x5 + 12*x1*x6 + 24*x1*x7 + 42*x1*x8 + 84*x1*x9 + 2*x2^2 + 4*x2*x3 + 8*x2*x4 + 12*x2*x5 + 24*x2*x6 + 42*x2*x7 + 84*x2*x8 + 156*x2*x9 + 4*x3^2 + 12*x3*x4 + 24*x3*x5 + 42*x3*x6 + 84*x3*x7 + 156*x3*x8 + 312*x3*x9 + 12*x4^2 + 42*x4*x5 + 84*x4*x6 + 156*x4*x7 + 312*x4*x8 + 596*x4*x9 + 42*x5^2 + 156*x5*x6 + 312*x5*x7 + 596*x5*x8 + 1192*x5*x9 + 156*x6^2 + 596*x6*x7 + 1192*x6*x8 + 2314*x6*x9 + 596*x7^2 + 2314*x7*x8 + 4630*x7*x9 + 2315*x8^2 + 9082*x8*x9 + 9095*x9^2
> B2;
[   2    1    2    2    4    6   12   21   42]
[   1    2    2    4    6   12   21   42   78]
[   2    2    4    6   12   21   42   78  156]
[   2    4    6   12   21   42   78  156  298]
[   4    6   12   21   42   78  156  298  596]
[   6   12   21   42   78  156  298  596 1157]
[  12   21   42   78  156  298  596 1157 2315]
[  21   42   78  156  298  596 1157 2315 4541]
[  42   78  156  298  596 1157 2315 4541 9095]
> v := RMatrixSpace(Rationals(),1,9)![  0 ,  5,   0, -13,   0,   7,   0,  -1,   0];
> v;
[  0   5   0 -13   0   7   0  -1   0]
> v*B2*Transpose(v);
[5]
> v;
[  0   5   0 -13   0   7   0  -1   0]
> s3 := ShortVectors(L,2);
> s2 := ShortVectors(L,2);
> s2;
[
<( 1  0 -1  0  0  0  0  0  0), 1>,
<(1 0 0 0 0 0 0 0 0), 1>,
<( 2  0 -4  0  1  0  0  0  0), 1>,
<( 3  0 -9  0  6  0 -1  0  0), 1>,
<(  4   0 -17   0  20   0  -8   0   1), 1>,
<(  4   3 -17  -9  20   6  -8  -1   1), 1>,
<(  4   5 -17 -13  20   7  -8  -1   1), 1>,
<(  4   7 -17 -14  20   7  -8  -1   1), 1>,
<(0 1 0 0 0 0 0 0 0), 1>,
<( 1 -1 -1  0  0  0  0  0  0), 2>,
<( 1 -1  0  0  0  0  0  0  0), 2>,
<( 1  0 -8  0 14  0 -7  0  1), 2>,
<( 1  0 -5  0  5  0 -1  0  0), 2>,
<( 1  0 -4  0  1  0  0  0  0), 2>,
<( 1  0 -3  0  1  0  0  0  0), 2>,
<( 1  1 -1  0  0  0  0  0  0), 2>,
<(1 1 0 0 0 0 0 0 0), 2>,
<( 1  3 -8 -9 14  6 -7 -1  1), 2>,
<(  1   5  -8 -13  14   7  -7  -1   1), 2>,
<(  1   7  -8 -14  14   7  -7  -1   1), 2>,
<( 2 -1 -4  0  1  0  0  0  0), 2>,
<(  2   0 -13   0  19   0  -8   0   1), 2>,
<( 2  0 -9  0  6  0 -1  0  0), 2>,
<( 2  0 -8  0  6  0 -1  0  0), 2>,
<( 2  0 -1  0  0  0  0  0  0), 2>,
<( 2  1 -4  0  1  0  0  0  0), 2>,
<(  2   3 -13  -9  19   6  -8  -1   1), 2>,
<(  2   5 -13 -13  19   7  -8  -1   1), 2>,
<(  2   7 -13 -14  19   7  -8  -1   1), 2>,
<( 3 -1 -9  0  6  0 -1  0  0), 2>,
<(  3   0 -17   0  20   0  -8   0   1), 2>,
<(  3   0 -16   0  20   0  -8   0   1), 2>,
<( 3  0 -5  0  1  0  0  0  0), 2>,
<( 3  0 -4  0  1  0  0  0  0), 2>,
<( 3  1 -9  0  6  0 -1  0  0), 2>,
<(  3   3 -17  -9  20   6  -8  -1   1), 2>,
<(  3   3 -16  -9  20   6  -8  -1   1), 2>,
<(  3   5 -17 -13  20   7  -8  -1   1), 2>,
<(  3   5 -16 -13  20   7  -8  -1   1), 2>,
<(  3   7 -17 -14  20   7  -8  -1   1), 2>,
<(  3   7 -16 -14  20   7  -8  -1   1), 2>,
<(  4  -1 -17   0  20   0  -8   0   1), 2>,
<(  4   0 -10   0   6   0  -1   0   0), 2>,
<( 4  0 -9  0  6  0 -1  0  0), 2>,
<(  4   1 -17   0  20   0  -8   0   1), 2>,
<(  4   2 -17  -9  20   6  -8  -1   1), 2>,
<(  4   4 -17 -13  20   7  -8  -1   1), 2>,
<(  4   4 -17  -9  20   6  -8  -1   1), 2>,
<(  4   6 -17 -14  20   7  -8  -1   1), 2>,
<(  4   6 -17 -13  20   7  -8  -1   1), 2>,
<(  4   8 -17 -14  20   7  -8  -1   1), 2>,
<(  5   0 -18   0  20   0  -8   0   1), 2>,
<(  5   0 -17   0  20   0  -8   0   1), 2>,
<(  5   0 -13   0   7   0  -1   0   0), 2>,
<(  5   3 -18  -9  20   6  -8  -1   1), 2>,
<(  5   3 -17  -9  20   6  -8  -1   1), 2>,
<(  5   5 -18 -13  20   7  -8  -1   1), 2>,
<(  5   5 -17 -13  20   7  -8  -1   1), 2>,
<(  5   7 -18 -14  20   7  -8  -1   1), 2>,
<(  5   7 -17 -14  20   7  -8  -1   1), 2>,
<(  6   0 -21   0  21   0  -8   0   1), 2>,
<(  6   3 -21  -9  21   6  -8  -1   1), 2>,
<(  6   5 -21 -13  21   7  -8  -1   1), 2>,
<(  6   7 -21 -14  21   7  -8  -1   1), 2>,
<(  7   0 -26   0  26   0  -9   0   1), 2>,
<(  7   3 -26  -9  26   6  -9  -1   1), 2>,
<(  7   5 -26 -13  26   7  -9  -1   1), 2>,
<(  7   7 -26 -14  26   7  -9  -1   1), 2>,
<(  8   3 -34  -9  40   6 -16  -1   2), 2>,
<(  8   5 -34 -13  40   7 -16  -1   2), 2>,
<(  8   7 -34 -14  40   7 -16  -1   2), 2>,
<(  8   8 -34 -22  40  13 -16  -2   2), 2>,
<(  8  10 -34 -23  40  13 -16  -2   2), 2>,
<(  8  12 -34 -27  40  14 -16  -2   2), 2>,
<( 0  2  0 -4  0  1  0  0  0), 2>,
<( 0  2  0 -1  0  0  0  0  0), 2>,
<( 0  3  0 -9  0  6  0 -1  0), 2>,
<( 0  4  0 -5  0  1  0  0  0), 2>,
<(  0   5   0 -13   0   7   0  -1   0), 2>,
<(  0   7   0 -14   0   7   0  -1   0), 2>,
<(0 0 1 0 0 0 0 0 0), 2>
]
> s2 := ShortVectors(L2,2);
> s2;
[
<( 2 -1 -8  4  8 -4 -2  1  0), 1>,
<( 1 -4  0  7 -3 -2  1  0  0), 2>,
<( 1 -3 -3  4  1 -1  0  0  0), 2>,
<( 1 -3  1  6 -3 -2  1  0  0), 2>,
<( 1 -2 -3  4  1 -1  0  0  0), 2>,
<( 1 -2 -2  3  1 -1  0  0  0), 2>,
<( 1 -2 -1  1  0  0  0  0  0), 2>,
<( 1 -2 -1  5 -2 -2  1  0  0), 2>,
<( 1 -1 -4  2  2 -1  0  0  0), 2>,
<( 1 -1 -2  1  0  0  0  0  0), 2>,
<( 1 -1 -2  3  1 -1  0  0  0), 2>,
<( 1 -1 -1  1  0  0  0  0  0), 2>,
<( 1 -1  0  0  0  0  0  0  0), 2>,
<( 1  0 -4 -2  4  1 -1  0  0), 2>,
<( 1  0 -4  2  2 -1  0  0  0), 2>,
<( 1  0 -2 -1  1  0  0  0  0), 2>,
<( 1  0 -2  1  0  0  0  0  0), 2>,
<( 1  0 -1  0  0  0  0  0  0), 2>,
<(1 0 0 0 0 0 0 0 0), 2>,
<( 1  1 -5 -2  4  1 -1  0  0), 2>,
<( 1  1 -3 -3  4  1 -1  0  0), 2>,
<( 1  1 -3 -1  1  0  0  0  0), 2>,
<( 1  1 -2 -1  1  0  0  0  0), 2>,
<( 1  1 -1  0  0  0  0  0  0), 2>,
<( 1  2 -5 -4  5  1 -1  0  0), 2>,
<( 1  2 -4 -3  4  1 -1  0  0), 2>,
<( 1  2 -3 -5  3  2 -1  0  0), 2>,
<( 1  2 -3 -1  1  0  0  0  0), 2>,
<( 1  3 -8 -9 14  6 -7 -1  1), 2>,
<( 1  3 -6 -4  5  1 -1  0  0), 2>,
<( 1  3 -2 -6  3  2 -1  0  0), 2>,
<(  1   4 -10 -10  15   6  -7  -1   1), 2>,
<( 1  4 -4 -7  4  2 -1  0  0), 2>,
<(  1   5  -9 -11  15   6  -7  -1   1), 2>,
<( 2 -3 -3  4  1 -1  0  0  0), 2>,
<( 2 -2 -5  3  2 -1  0  0  0), 2>,
<( 2 -2 -4  4  1 -1  0  0  0), 2>,
<( 2 -1 -6  3  2 -1  0  0  0), 2>,
<( 2 -1 -4  2  2 -1  0  0  0), 2>,
<( 2 -1 -2  1  0  0  0  0  0), 2>,
<( 2  0 -5 -2  4  1 -1  0  0), 2>,
<( 2  0 -5  2  2 -1  0  0  0), 2>,
<( 2  0 -4  0  1  0  0  0  0), 2>,
<( 2  1 -9 -4 12  4 -6 -1  1), 2>,
<( 2  1 -7 -3  5  1 -1  0  0), 2>,
<( 2  1 -5 -2  4  1 -1  0  0), 2>,
<( 2  1 -3 -1  1  0  0  0  0), 2>,
<(  2   2 -12  -7  16   5  -7  -1   1), 2>,
<( 2  2 -7 -3  5  1 -1  0  0), 2>,
<( 2  2 -6 -4  5  1 -1  0  0), 2>,
<(  2   3 -12  -7  16   5  -7  -1   1), 2>,
<(  2   3 -10 -10  15   6  -7  -1   1), 2>,
<( 2  3 -6 -4  5  1 -1  0  0), 2>,
<(  2   4 -11 -10  15   6  -7  -1   1), 2>,
<(  2   4 -10 -10  15   6  -7  -1   1), 2>,
<(  2   5 -13 -13  19   7  -8  -1   1), 2>,
<(  2   5 -11 -10  15   6  -7  -1   1), 2>,
<(  2   6 -14 -13  19   7  -8  -1   1), 2>,
<(  2   7 -12 -16  18   8  -8  -1   1), 2>,
<( 3 -2 -6  3  2 -1  0  0  0), 2>,
<( 3 -1 -6  3  2 -1  0  0  0), 2>,
<( 3  0 -9  0  6  0 -1  0  0), 2>,
<(  3   1 -13  -6  16   5  -7  -1   1), 2>,
<( 3  1 -7 -3  5  1 -1  0  0), 2>,
<(  3   2 -14  -6  16   5  -7  -1   1), 2>,
<(  3   2 -12  -7  16   5  -7  -1   1), 2>,
<( 3  2 -8 -3  5  1 -1  0  0), 2>,
<(  3   3 -14  -8  17   5  -7  -1   1), 2>,
<(  3   3 -13  -7  16   5  -7  -1   1), 2>,
<(  3   3 -12  -9  15   6  -7  -1   1), 2>,
<(  3   4 -15 -12  19   7  -8  -1   1), 2>,
<(  3   4 -15  -8  17   5  -7  -1   1), 2>,
<(  3   4 -11 -10  15   6  -7  -1   1), 2>,
<(  3   5 -15 -12  19   7  -8  -1   1), 2>,
<(  3   5 -14 -13  19   7  -8  -1   1), 2>,
<(  3   5 -13 -11  16   6  -7  -1   1), 2>,
<(  3   6 -16 -14  20   7  -8  -1   1), 2>,
<(  3   6 -14 -13  19   7  -8  -1   1), 2>,
<(  3   7 -16 -14  20   7  -8  -1   1), 2>,
<(  4   1 -14  -6  16   5  -7  -1   1), 2>,
<(  4   2 -16  -7  17   5  -7  -1   1), 2>,
<(  4   2 -14  -6  16   5  -7  -1   1), 2>,
<(  4   3 -17  -9  20   6  -8  -1   1), 2>,
<(  4   3 -16  -7  17   5  -7  -1   1), 2>,
<(  4   3 -15  -8  17   5  -7  -1   1), 2>,
<(  4   4 -19 -10  21   6  -8  -1   1), 2>,
<(  4   4 -15 -12  19   7  -8  -1   1), 2>,
<(  4   4 -15  -8  17   5  -7  -1   1), 2>,
<(  4   5 -18 -11  21   6  -8  -1   1), 2>,
<(  4   5 -17 -13  20   7  -8  -1   1), 2>,
<(  4   5 -16 -12  19   7  -8  -1   1), 2>,
<(  4   6 -18 -13  20   7  -8  -1   1), 2>,
<(  4   6 -16 -14  20   7  -8  -1   1), 2>,
<(  4   7 -17 -14  20   7  -8  -1   1), 2>,
<(  5   1 -18  -4  18   4  -7  -1   1), 2>,
<(  5   2 -16  -7  17   5  -7  -1   1), 2>,
<(  5   3 -19 -10  21   6  -8  -1   1), 2>,
<(  5   3 -17  -7  17   5  -7  -1   1), 2>,
<(  5   4 -20 -10  21   6  -8  -1   1), 2>,
<(  5   4 -19 -10  21   6  -8  -1   1), 2>,
<(  5   5 -20 -10  21   6  -8  -1   1), 2>,
<(  5   5 -18 -13  20   7  -8  -1   1), 2>,
<(  5   6 -18 -13  20   7  -8  -1   1), 2>,
<(  5   7 -21 -16  24   8  -9  -1   1), 2>,
<(  6   3 -21  -9  21   6  -8  -1   1), 2>,
<(  6   4 -20 -10  21   6  -8  -1   1), 2>,
<(  6   5 -22 -11  22   6  -8  -1   1), 2>,
<(  7   8 -30 -20  36  12 -15  -2   2), 2>,
<( 0  1 -3 -3  4  1 -1  0  0), 2>,
<( 0  1 -2 -1  1  0  0  0  0), 2>,
<( 0  1 -1  0  0  0  0  0  0), 2>,
<(0 1 0 0 0 0 0 0 0), 2>,
<( 0  1  1 -1  0  0  0  0  0), 2>,
<( 0  1  2 -3 -1  1  0  0  0), 2>,
<( 0  2 -3 -3  4  1 -1  0  0), 2>,
<( 0  2 -1 -6  3  2 -1  0  0), 2>,
<( 0  2 -1 -2  1  0  0  0  0), 2>,
<( 0  2  1 -3 -1  1  0  0  0), 2>,
<( 0  3 -2 -6  3  2 -1  0  0), 2>,
<( 0  3 -1 -6  3  2 -1  0  0), 2>,
<( 0  4 -2 -6  3  2 -1  0  0), 2>
]
> s3 := ShortVectors(L2,3);
> s3;
[
<( 2 -1 -8  4  8 -4 -2  1  0), 1>,
<( 1 -4  0  7 -3 -2  1  0  0), 2>,
<( 1 -3 -3  4  1 -1  0  0  0), 2>,
<( 1 -3  1  6 -3 -2  1  0  0), 2>,
<( 1 -2 -3  4  1 -1  0  0  0), 2>,
<( 1 -2 -2  3  1 -1  0  0  0), 2>,
<( 1 -2 -1  1  0  0  0  0  0), 2>,
<( 1 -2 -1  5 -2 -2  1  0  0), 2>,
<( 1 -1 -4  2  2 -1  0  0  0), 2>,
<( 1 -1 -2  1  0  0  0  0  0), 2>,
<( 1 -1 -2  3  1 -1  0  0  0), 2>,
<( 1 -1 -1  1  0  0  0  0  0), 2>,
<( 1 -1  0  0  0  0  0  0  0), 2>,
<( 1  0 -4 -2  4  1 -1  0  0), 2>,
<( 1  0 -4  2  2 -1  0  0  0), 2>,
<( 1  0 -2 -1  1  0  0  0  0), 2>,
<( 1  0 -2  1  0  0  0  0  0), 2>,
<( 1  0 -1  0  0  0  0  0  0), 2>,
<(1 0 0 0 0 0 0 0 0), 2>,
<( 1  1 -5 -2  4  1 -1  0  0), 2>,
<( 1  1 -3 -3  4  1 -1  0  0), 2>,
<( 1  1 -3 -1  1  0  0  0  0), 2>,
<( 1  1 -2 -1  1  0  0  0  0), 2>,
<( 1  1 -1  0  0  0  0  0  0), 2>,
<( 1  2 -5 -4  5  1 -1  0  0), 2>,
<( 1  2 -4 -3  4  1 -1  0  0), 2>,
<( 1  2 -3 -5  3  2 -1  0  0), 2>,
<( 1  2 -3 -1  1  0  0  0  0), 2>,
<( 1  3 -8 -9 14  6 -7 -1  1), 2>,
<( 1  3 -6 -4  5  1 -1  0  0), 2>,
<( 1  3 -2 -6  3  2 -1  0  0), 2>,
<(  1   4 -10 -10  15   6  -7  -1   1), 2>,
<( 1  4 -4 -7  4  2 -1  0  0), 2>,
<(  1   5  -9 -11  15   6  -7  -1   1), 2>,
<( 2 -3 -3  4  1 -1  0  0  0), 2>,
<( 2 -2 -5  3  2 -1  0  0  0), 2>,
<( 2 -2 -4  4  1 -1  0  0  0), 2>,
<( 2 -1 -6  3  2 -1  0  0  0), 2>,
<( 2 -1 -4  2  2 -1  0  0  0), 2>,
<( 2 -1 -2  1  0  0  0  0  0), 2>,
<( 2  0 -5 -2  4  1 -1  0  0), 2>,
<( 2  0 -5  2  2 -1  0  0  0), 2>,
<( 2  0 -4  0  1  0  0  0  0), 2>,
<( 2  1 -9 -4 12  4 -6 -1  1), 2>,
<( 2  1 -7 -3  5  1 -1  0  0), 2>,
<( 2  1 -5 -2  4  1 -1  0  0), 2>,
<( 2  1 -3 -1  1  0  0  0  0), 2>,
<(  2   2 -12  -7  16   5  -7  -1   1), 2>,
<( 2  2 -7 -3  5  1 -1  0  0), 2>,
<( 2  2 -6 -4  5  1 -1  0  0), 2>,
<(  2   3 -12  -7  16   5  -7  -1   1), 2>,
<(  2   3 -10 -10  15   6  -7  -1   1), 2>,
<( 2  3 -6 -4  5  1 -1  0  0), 2>,
<(  2   4 -11 -10  15   6  -7  -1   1), 2>,
<(  2   4 -10 -10  15   6  -7  -1   1), 2>,
<(  2   5 -13 -13  19   7  -8  -1   1), 2>,
<(  2   5 -11 -10  15   6  -7  -1   1), 2>,
<(  2   6 -14 -13  19   7  -8  -1   1), 2>,
<(  2   7 -12 -16  18   8  -8  -1   1), 2>,
<( 3 -2 -6  3  2 -1  0  0  0), 2>,
<( 3 -1 -6  3  2 -1  0  0  0), 2>,
<( 3  0 -9  0  6  0 -1  0  0), 2>,
<(  3   1 -13  -6  16   5  -7  -1   1), 2>,
<( 3  1 -7 -3  5  1 -1  0  0), 2>,
<(  3   2 -14  -6  16   5  -7  -1   1), 2>,
<(  3   2 -12  -7  16   5  -7  -1   1), 2>,
<( 3  2 -8 -3  5  1 -1  0  0), 2>,
<(  3   3 -14  -8  17   5  -7  -1   1), 2>,
<(  3   3 -13  -7  16   5  -7  -1   1), 2>,
<(  3   3 -12  -9  15   6  -7  -1   1), 2>,
<(  3   4 -15 -12  19   7  -8  -1   1), 2>,
<(  3   4 -15  -8  17   5  -7  -1   1), 2>,
<(  3   4 -11 -10  15   6  -7  -1   1), 2>,
<(  3   5 -15 -12  19   7  -8  -1   1), 2>,
<(  3   5 -14 -13  19   7  -8  -1   1), 2>,
<(  3   5 -13 -11  16   6  -7  -1   1), 2>,
<(  3   6 -16 -14  20   7  -8  -1   1), 2>,
<(  3   6 -14 -13  19   7  -8  -1   1), 2>,
<(  3   7 -16 -14  20   7  -8  -1   1), 2>,
<(  4   1 -14  -6  16   5  -7  -1   1), 2>,
<(  4   2 -16  -7  17   5  -7  -1   1), 2>,
<(  4   2 -14  -6  16   5  -7  -1   1), 2>,
<(  4   3 -17  -9  20   6  -8  -1   1), 2>,
<(  4   3 -16  -7  17   5  -7  -1   1), 2>,
<(  4   3 -15  -8  17   5  -7  -1   1), 2>,
<(  4   4 -19 -10  21   6  -8  -1   1), 2>,
<(  4   4 -15 -12  19   7  -8  -1   1), 2>,
<(  4   4 -15  -8  17   5  -7  -1   1), 2>,
<(  4   5 -18 -11  21   6  -8  -1   1), 2>,
<(  4   5 -17 -13  20   7  -8  -1   1), 2>,
<(  4   5 -16 -12  19   7  -8  -1   1), 2>,
<(  4   6 -18 -13  20   7  -8  -1   1), 2>,
<(  4   6 -16 -14  20   7  -8  -1   1), 2>,
<(  4   7 -17 -14  20   7  -8  -1   1), 2>,
<(  5   1 -18  -4  18   4  -7  -1   1), 2>,
<(  5   2 -16  -7  17   5  -7  -1   1), 2>,
<(  5   3 -19 -10  21   6  -8  -1   1), 2>,
<(  5   3 -17  -7  17   5  -7  -1   1), 2>,
<(  5   4 -20 -10  21   6  -8  -1   1), 2>,
<(  5   4 -19 -10  21   6  -8  -1   1), 2>,
<(  5   5 -20 -10  21   6  -8  -1   1), 2>,
<(  5   5 -18 -13  20   7  -8  -1   1), 2>,
<(  5   6 -18 -13  20   7  -8  -1   1), 2>,
<(  5   7 -21 -16  24   8  -9  -1   1), 2>,
<(  6   3 -21  -9  21   6  -8  -1   1), 2>,
<(  6   4 -20 -10  21   6  -8  -1   1), 2>,
<(  6   5 -22 -11  22   6  -8  -1   1), 2>,
<(  7   8 -30 -20  36  12 -15  -2   2), 2>,
<( 0  1 -3 -3  4  1 -1  0  0), 2>,
<( 0  1 -2 -1  1  0  0  0  0), 2>,
<( 0  1 -1  0  0  0  0  0  0), 2>,
<(0 1 0 0 0 0 0 0 0), 2>,
<( 0  1  1 -1  0  0  0  0  0), 2>,
<( 0  1  2 -3 -1  1  0  0  0), 2>,
<( 0  2 -3 -3  4  1 -1  0  0), 2>,
<( 0  2 -1 -6  3  2 -1  0  0), 2>,
<( 0  2 -1 -2  1  0  0  0  0), 2>,
<( 0  2  1 -3 -1  1  0  0  0), 2>,
<( 0  3 -2 -6  3  2 -1  0  0), 2>,
<( 0  3 -1 -6  3  2 -1  0  0), 2>,
<( 0  4 -2 -6  3  2 -1  0  0), 2>,
<(  1  -6   1  15  -7 -10   5   2  -1), 3>,
<( 1 -5 -4 11  4 -6 -1  1  0), 3>,
<(  1  -5   2  14  -7 -10   5   2  -1), 3>,
<( 1 -4 -6 10  5 -6 -1  1  0), 3>,
<( 1 -4 -2  8  3 -5 -1  1  0), 3>,
<(  1  -4   0  13  -6 -10   5   2  -1), 3>,
<( 1 -3 -5  5  7 -4 -2  1  0), 3>,
<( 1 -3 -5  9  5 -6 -1  1  0), 3>,
<( 1 -3 -4  7  4 -5 -1  1  0), 3>,
<( 1 -3 -3  8  3 -5 -1  1  0), 3>,
<( 1 -2 -7  4  8 -4 -2  1  0), 3>,
<( 1 -2 -6  5  7 -4 -2  1  0), 3>,
<( 1 -2 -5  5  7 -4 -2  1  0), 3>,
<( 1 -2 -5  7  4 -5 -1  1  0), 3>,
<( 1 -2 -3  6  4 -5 -1  1  0), 3>,
<( 1 -1 -8  4  8 -4 -2  1  0), 3>,
<( 1 -1 -7  4  8 -4 -2  1  0), 3>,
<( 1 -1 -6  3  8 -4 -2  1  0), 3>,
<( 1 -1 -6  5  7 -4 -2  1  0), 3>,
<( 1 -1 -4  2  6 -3 -2  1  0), 3>,
<( 1 -1 -4  6  4 -5 -1  1  0), 3>,
<( 1 -1  2 -1 -6  3  2 -1  0), 3>,
<( 1  0 -8  4  8 -4 -2  1  0), 3>,
<( 1  0 -7  3  8 -4 -2  1  0), 3>,
<( 1  0 -6  1  7 -3 -2  1  0), 3>,
<( 1  0 -6  3  8 -4 -2  1  0), 3>,
<( 1  0 -4  2  6 -3 -2  1  0), 3>,
<( 1  0  2 -1 -6  3  2 -1  0), 3>,
<( 1  1 -7 -1 10 -2 -3  1  0), 3>,
<( 1  1 -7  3  8 -4 -2  1  0), 3>,
<( 1  1 -6  1  7 -3 -2  1  0), 3>,
<( 1  1 -5  0  7 -3 -2  1  0), 3>,
<( 1  1 -1 -4 -2  4  1 -1  0), 3>,
<( 1  2 -9 -2 11 -2 -3  1  0), 3>,
<(  1   2  -5 -10   8   9  -5  -2   1), 3>,
<( 1  2 -5  0  7 -3 -2  1  0), 3>,
<( 1  2  1 -7 -3  5  1 -1  0), 3>,
<( 1  3 -8 -3 11 -2 -3  1  0), 3>,
<(  1   3  -6 -10   8   9  -5  -2   1), 3>,
<(  1   3  -4 -11   8   9  -5  -2   1), 3>,
<( 1  3  0 -7 -3  5  1 -1  0), 3>,
<(  1   4  -6 -12   9   9  -5  -2   1), 3>,
<(  1   4  -5 -11   8   9  -5  -2   1), 3>,
<(  1   4  -4 -13   7  10  -5  -2   1), 3>,
<(  1   5  -7 -16  11  11  -6  -2   1), 3>,
<(  1   5  -7 -12   9   9  -5  -2   1), 3>,
<(  1   5  -3 -14   7  10  -5  -2   1), 3>,
<(  1   6  -7 -16  11  11  -6  -2   1), 3>,
<(  1   6  -6 -17  11  11  -6  -2   1), 3>,
<(  1   6  -5 -15   8  10  -5  -2   1), 3>,
<(  1   7  -8 -18  12  11  -6  -2   1), 3>,
<(  1   7  -6 -17  11  11  -6  -2   1), 3>,
<(  1   8  -8 -18  12  11  -6  -2   1), 3>,
<( 2 -5 -6 10  5 -6 -1  1  0), 3>,
<( 2 -4 -7 10  5 -6 -1  1  0), 3>,
<( 2 -4 -6 10  5 -6 -1  1  0), 3>,
<( 2 -3 -9  7  9 -5 -2  1  0), 3>,
<( 2 -3 -7  6  7 -4 -2  1  0), 3>,
<( 2 -3 -7 10  5 -6 -1  1  0), 3>,
<( 2 -3 -5  7  4 -5 -1  1  0), 3>,
<(  2  -2 -10   7   9  -5  -2   1   0), 3>,
<( 2 -2 -9  5  8 -4 -2  1  0), 3>,
<( 2 -2 -8  4  8 -4 -2  1  0), 3>,
<( 2 -2 -7  4  8 -4 -2  1  0), 3>,
<( 2 -2 -6  5  7 -4 -2  1  0), 3>,
<( 2 -2 -5  7  4 -5 -1  1  0), 3>,
<(  2   0 -11   1  12  -3  -3   1   0), 3>,
<(  2   0 -10   3   9  -4  -2   1   0), 3>,
<( 2  0 -9  4  8 -4 -2  1  0), 3>,
<( 2  0 -8  4  8 -4 -2  1  0), 3>,
<( 2  0 -7  3  8 -4 -2  1  0), 3>,
<( 2  0 -6  1  7 -3 -2  1  0), 3>,
<(  2   1 -11   1  12  -3  -3   1   0), 3>,
<( 2  1 -9 -2 11 -2 -3  1  0), 3>,
<( 2  1 -9  2  9 -4 -2  1  0), 3>,
<( 2  1 -7  1  7 -3 -2  1  0), 3>,
<(  2   2 -10  -2  11  -2  -3   1   0), 3>,
<( 2  2 -9 -2 11 -2 -3  1  0), 3>,
<(  2   2  -6 -10   8   9  -5  -2   1), 3>,
<(  2   3 -10  -2  11  -2  -3   1   0), 3>,
<(  2   3  -8 -11   9   9  -5  -2   1), 3>,
<(  2   3  -6 -10   8   9  -5  -2   1), 3>,
<(  2   4  -9 -13  12  10  -6  -2   1), 3>,
<(  2   4  -8 -11   9   9  -5  -2   1), 3>,
<(  2   4  -7 -12   9   9  -5  -2   1), 3>,
<(  2   5 -11 -14  13  10  -6  -2   1), 3>,
<(  2   5  -7 -16  11  11  -6  -2   1), 3>,
<(  2   5  -7 -12   9   9  -5  -2   1), 3>,
<(  2   6 -10 -15  13  10  -6  -2   1), 3>,
<(  2   6  -9 -17  12  11  -6  -2   1), 3>,
<(  2   6  -8 -16  11  11  -6  -2   1), 3>,
<(  2   7 -10 -17  12  11  -6  -2   1), 3>,
<(  2   7  -8 -18  12  11  -6  -2   1), 3>,
<(  2   8  -9 -18  12  11  -6  -2   1), 3>,
<( 3 -5 -8 11  5 -6 -1  1  0), 3>,
<(  3  -4 -11   8   9  -5  -2   1   0), 3>,
<( 3 -4 -7 10  5 -6 -1  1  0), 3>,
<(  3  -3 -11   8   9  -5  -2   1   0), 3>,
<(  3  -3 -10   7   9  -5  -2   1   0), 3>,
<( 3 -3 -9  5  8 -4 -2  1  0), 3>,
<( 3 -3 -9  9  6 -6 -1  1  0), 3>,
<(  3  -2 -12   6  10  -5  -2   1   0), 3>,
<(  3  -2 -10   5   8  -4  -2   1   0), 3>,
<(  3  -2 -10   7   9  -5  -2   1   0), 3>,
<( 3 -2 -9  5  8 -4 -2  1  0), 3>,
<( 3 -2 -8  4  8 -4 -2  1  0), 3>,
<(  3  -1 -12   2  12  -3  -3   1   0), 3>,
<(  3  -1 -12   6  10  -5  -2   1   0), 3>,
<(  3  -1 -10   3   9  -4  -2   1   0), 3>,
<(  3  -1 -10   5   8  -4  -2   1   0), 3>,
<( 3 -1 -9  4  8 -4 -2  1  0), 3>,
<( 3 -1 -8  4  8 -4 -2  1  0), 3>,
<(  3   0 -13   2  12  -3  -3   1   0), 3>,
<(  3   0 -11   1  12  -3  -3   1   0), 3>,
<(  3   0 -11   3   9  -4  -2   1   0), 3>,
<(  3   0 -10   3   9  -4  -2   1   0), 3>,
<( 3  0 -9  4  8 -4 -2  1  0), 3>,
<(  3   1 -13   0  13  -3  -3   1   0), 3>,
<(  3   1 -12   1  12  -3  -3   1   0), 3>,
<(  3   1 -11  -1  11  -2  -3   1   0), 3>,
<(  3   1 -11   3   9  -4  -2   1   0), 3>,
<(  3   2 -16  -5  22   2  -9   0   1), 3>,
<(  3   2 -14   0  13  -3  -3   1   0), 3>,
<(  3   2 -10  -8  10   8  -5  -2   1), 3>,
<(  3   2 -10  -2  11  -2  -3   1   0), 3>,
<(  3   3 -18  -6  23   2  -9   0   1), 3>,
<(  3   3 -12  -3  12  -2  -3   1   0), 3>,
<(  3   3  -8 -11   9   9  -5  -2   1), 3>,
<(  3   4 -17  -7  23   2  -9   0   1), 3>,
<(  3   4 -11 -14  13  10  -6  -2   1), 3>,
<(  3   4  -9 -11   9   9  -5  -2   1), 3>,
<(  3   5 -12 -14  13  10  -6  -2   1), 3>,
<(  3   5 -11 -14  13  10  -6  -2   1), 3>,
<(  3   6 -12 -14  13  10  -6  -2   1), 3>,
<(  3   6 -10 -17  12  11  -6  -2   1), 3>,
<(  3   7 -10 -17  12  11  -6  -2   1), 3>,
<(  3   8 -13 -20  16  12  -7  -2   1), 3>,
<(  4  -4 -11   8   9  -5  -2   1   0), 3>,
<(  4  -3 -13   7  10  -5  -2   1   0), 3>,
<(  4  -3 -12   8   9  -5  -2   1   0), 3>,
<(  4  -2 -14   7  10  -5  -2   1   0), 3>,
<(  4  -2 -12   6  10  -5  -2   1   0), 3>,
<(  4  -2 -10   5   8  -4  -2   1   0), 3>,
<(  4  -1 -13   2  12  -3  -3   1   0), 3>,
<(  4  -1 -13   6  10  -5  -2   1   0), 3>,
<(  4  -1 -12   4   9  -4  -2   1   0), 3>,
<(  4   0 -17   0  20   0  -8   0   1), 3>,
<(  4   0 -15   1  13  -3  -3   1   0), 3>,
<(  4   0 -13   2  12  -3  -3   1   0), 3>,
<(  4   0 -11   3   9  -4  -2   1   0), 3>,
<(  4   1 -20  -3  24   1  -9   0   1), 3>,
<(  4   1 -15   1  13  -3  -3   1   0), 3>,
<(  4   1 -14   0  13  -3  -3   1   0), 3>,
<(  4   2 -20  -3  24   1  -9   0   1), 3>,
<(  4   2 -18  -6  23   2  -9   0   1), 3>,
<(  4   2 -14   0  13  -3  -3   1   0), 3>,
<(  4   3 -19  -6  23   2  -9   0   1), 3>,
<(  4   3 -18  -6  23   2  -9   0   1), 3>,
<(  4   4 -21  -9  27   3 -10   0   1), 3>,
<(  4   4 -19  -6  23   2  -9   0   1), 3>,
<(  4   4 -13 -13  13  10  -6  -2   1), 3>,
<(  4   5 -22  -9  27   3 -10   0   1), 3>,
<(  4   5 -12 -14  13  10  -6  -2   1), 3>,
<(  4   6 -20 -12  26   4 -10   0   1), 3>,
<(  4   6 -14 -15  14  10  -6  -2   1), 3>,
<(  5  -3 -14   7  10  -5  -2   1   0), 3>,
<(  5  -2 -14   7  10  -5  -2   1   0), 3>,
<(  5  -1 -17   4  14  -4  -3   1   0), 3>,
<(  5   0 -21  -2  24   1  -9   0   1), 3>,
<(  5   0 -15   1  13  -3  -3   1   0), 3>,
<(  5   1 -22  -2  24   1  -9   0   1), 3>,
<(  5   1 -20  -3  24   1  -9   0   1), 3>,
<(  5   1 -16   1  13  -3  -3   1   0), 3>,
<(  5   2 -22  -4  25   1  -9   0   1), 3>,
<(  5   2 -21  -3  24   1  -9   0   1), 3>,
<(  5   2 -20  -5  23   2  -9   0   1), 3>,
<(  5   3 -23  -8  27   3 -10   0   1), 3>,
<(  5   3 -23  -4  25   1  -9   0   1), 3>,
<(  5   3 -19  -6  23   2  -9   0   1), 3>,
<(  5   4 -23  -8  27   3 -10   0   1), 3>,
<(  5   4 -22  -9  27   3 -10   0   1), 3>,
<(  5   4 -21  -7  24   2  -9   0   1), 3>,
<(  5   5 -24 -10  28   3 -10   0   1), 3>,
<(  5   5 -22  -9  27   3 -10   0   1), 3>,
<(  5   6 -24 -10  28   3 -10   0   1), 3>,
<(  5   9 -22 -24  28  16 -13  -3   2), 3>,
<(  6   0 -22  -2  24   1  -9   0   1), 3>,
<(  6   1 -24  -3  25   1  -9   0   1), 3>,
<(  6   1 -22  -2  24   1  -9   0   1), 3>,
<(  6   2 -25  -5  28   2 -10   0   1), 3>,
<(  6   2 -24  -3  25   1  -9   0   1), 3>,
<(  6   2 -23  -4  25   1  -9   0   1), 3>,
<(  6   3 -27  -6  29   2 -10   0   1), 3>,
<(  6   3 -23  -8  27   3 -10   0   1), 3>,
<(  6   3 -23  -4  25   1  -9   0   1), 3>,
<(  6   4 -26  -7  29   2 -10   0   1), 3>,
<(  6   4 -25  -9  28   3 -10   0   1), 3>,
<(  6   4 -24  -8  27   3 -10   0   1), 3>,
<(  6   5 -26  -9  28   3 -10   0   1), 3>,
<(  6   5 -24 -10  28   3 -10   0   1), 3>,
<(  6   6 -25 -10  28   3 -10   0   1), 3>,
<(  7   0 -26   0  26   0  -9   0   1), 3>,
<(  7   1 -24  -3  25   1  -9   0   1), 3>,
<(  7   2 -27  -6  29   2 -10   0   1), 3>,
<(  7   2 -25  -3  25   1  -9   0   1), 3>,
<(  7   3 -28  -6  29   2 -10   0   1), 3>,
<(  7   3 -27  -6  29   2 -10   0   1), 3>,
<(  7   4 -28  -6  29   2 -10   0   1), 3>,
<(  7   4 -26  -9  28   3 -10   0   1), 3>,
<(  7   5 -26  -9  28   3 -10   0   1), 3>,
<(  7   6 -29 -12  32   4 -11   0   1), 3>,
<(  8   2 -29  -5  29   2 -10   0   1), 3>,
<(  8   3 -28  -6  29   2 -10   0   1), 3>,
<(  8   4 -30  -7  30   2 -10   0   1), 3>,
<(  9   7 -38 -16  44   8 -17  -1   2), 3>,
<( 0  1 -4  0  7 -3 -2  1  0), 3>,
<( 0  1 -3  1  6 -3 -2  1  0), 3>,
<( 0  1  3 -6 -4  5  1 -1  0), 3>,
<( 0  1  3 -2 -6  3  2 -1  0), 3>,
<( 0  1  4 -4 -7  4  2 -1  0), 3>,
<( 0  2 -5  0  7 -3 -2  1  0), 3>,
<( 0  2 -1 -8  4  8 -4 -2  1), 3>,
<( 0  2  1 -7 -3  5  1 -1  0), 3>,
<( 0  2  3 -6 -4  5  1 -1  0), 3>,
<( 0  2  5 -5 -7  4  2 -1  0), 3>,
<(  0   3  -4 -11   8   9  -5  -2   1), 3>,
<( 0  3  1 -7 -3  5  1 -1  0), 3>,
<( 0  3  2 -8 -3  5  1 -1  0), 3>,
<(  0   4  -4 -11   8   9  -5  -2   1), 3>,
<(  0   4  -2 -14   7  10  -5  -2   1), 3>,
<( 0  4  2 -8 -3  5  1 -1  0), 3>,
<(  0   5  -3 -14   7  10  -5  -2   1), 3>,
<(  0   5  -2 -14   7  10  -5  -2   1), 3>,
<(  0   6  -5 -17  11  11  -6  -2   1), 3>,
<(  0   6  -3 -14   7  10  -5  -2   1), 3>,
<(  0   7  -6 -17  11  11  -6  -2   1), 3>,
<(  0   8  -4 -20  10  12  -6  -2   1), 3>,
<( 0  0  2 -1 -6  3  2 -1  0), 3>,
<( 0  0  4 -2 -6  3  2 -1  0), 3>,
<( 0  0  6 -3 -8  4  2 -1  0), 3>
]
> v := RMatrixSpace(Rationals(),1,9)![0,0,6,-3,-8,4,2,-1,0];
> v*B2*Transpose(v);
[3]
> B2;
[   2    1    2    2    4    6   12   21   42]
[   1    2    2    4    6   12   21   42   78]
[   2    2    4    6   12   21   42   78  156]
[   2    4    6   12   21   42   78  156  298]
[   4    6   12   21   42   78  156  298  596]
[   6   12   21   42   78  156  298  596 1157]
[  12   21   42   78  156  298  596 1157 2315]
[  21   42   78  156  298  596 1157 2315 4541]
[  42   78  156  298  596 1157 2315 4541 9095]
> #s3;
361
> #ShortVectors(L2,3) - #ShortVectors(L2,2);
240
>