let s^2=-2;
let Z^3=-Z^2-2*Z+2;
off nat;
phiZ:=(2*(s-1)+(3*s-2)*Z+s*Z^2)/4$
phi2Z:=(2*(-s-1)-(3*s+2)*Z-s*Z^2)/4$

% The following are zero:
Z^3+Z^2+2*Z-2;
phiZ^3+phiZ^2+2*phiZ-2;
phi2Z^3+phi2Z^2+2*phi2Z-2;
sub(Z=phiZ,phiZ)-phi2Z;
sub(X=phiZ,X^3+X^2+2*X-2);
sub(X=phi2Z,X^3+X^2+2*X-2);

ID3:=mat((1,0,0),
         (0,1,0),
         (0,0,1))$

% We embed the division algebra $\cD$ into the space of $3\times3$
% matrices over $m=\Q(s,Z)$, mapping $\sigma$ to the matrix $SIGMAT$,
% and $Z$ to the matrix $ZMAT$.

ZMAT:=mat((Z,   0,    0),
          (0,phiZ,    0),
          (0,   0,phi2Z))$

SIGMAT:=mat((0,1,0),
            (0,0,1),
            (3,0,0))$

gendt:=(1+2*s)/3$
gendti:=(1-2*s)/3$

FF:=mat((3,0,0),
        (0,0,1),
        (0,1,0))$

;end;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=2,p=3,\emptyset)$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

a1:=0/20$
a2:=-2/20$
a3:=4/20$
a4:=-14/20$
a5:=8/20$
a6:=0/20$
a7:=-2/20$
a8:=2/20$
a9:=4/20$
a10:=-2/20$
a11:=-4/20$
a12:=-4/20$
a13:=2/20$
a14:=-3/20$
a15:=2/20$
a16:=4/20$
a17:=-3/20$
a18:=2/20$

b1:=6/60$
b2:=18/60$
b3:=2/60$
b4:=0/60$
b5:=-6/60$
b6:=-2/60$
b7:=6/60$
b8:=0/60$
b9:=-4/60$
b10:=-24/60$
b11:=6/60$
b12:=-14/60$
b13:=-27/60$
b14:=-9/60$
b15:=-7/60$
b16:=-15/60$
b17:=9/60$
b18:=-5/60$

c1:=0/2$
c2:=0/2$
c3:=0/2$
c4:=-2/2$
c5:=0/2$
c6:=0/2$
c7:=0/2$
c8:=0/2$
c9:=0/2$
c10:=0/2$
c11:=0/2$
c12:=0/2$
c13:=1/2$
c14:=0/2$
c15:=0/2$
c16:=1/2$
c17:=0/2$
c18:=0/2$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

AA:=a1*ZMAT^0*SIGMAT^0+a2*ZMAT^0*SIGMAT^1+a3*ZMAT^0*SIGMAT^2+
a4*ZMAT^1*SIGMAT^0+a5*ZMAT^1*SIGMAT^1+a6*ZMAT^1*SIGMAT^2+
a7*ZMAT^2*SIGMAT^0+a8*ZMAT^2*SIGMAT^1+a9*ZMAT^2*SIGMAT^2+
a10*s*ZMAT^0*SIGMAT^0+a11*s*ZMAT^0*SIGMAT^1+a12*s*ZMAT^0*SIGMAT^2+
a13*s*ZMAT^1*SIGMAT^0+a14*s*ZMAT^1*SIGMAT^1+a15*s*ZMAT^1*SIGMAT^2+
a16*s*ZMAT^2*SIGMAT^0+a17*s*ZMAT^2*SIGMAT^1+a18*s*ZMAT^2*SIGMAT^2$

BB:=b1*ZMAT^0*SIGMAT^0+b2*ZMAT^0*SIGMAT^1+b3*ZMAT^0*SIGMAT^2+
b4*ZMAT^1*SIGMAT^0+b5*ZMAT^1*SIGMAT^1+b6*ZMAT^1*SIGMAT^2+
b7*ZMAT^2*SIGMAT^0+b8*ZMAT^2*SIGMAT^1+b9*ZMAT^2*SIGMAT^2+
b10*s*ZMAT^0*SIGMAT^0+b11*s*ZMAT^0*SIGMAT^1+b12*s*ZMAT^0*SIGMAT^2+
b13*s*ZMAT^1*SIGMAT^0+b14*s*ZMAT^1*SIGMAT^1+b15*s*ZMAT^1*SIGMAT^2+
b16*s*ZMAT^2*SIGMAT^0+b17*s*ZMAT^2*SIGMAT^1+b18*s*ZMAT^2*SIGMAT^2$

CC:=c1*ZMAT^0*SIGMAT^0+c2*ZMAT^0*SIGMAT^1+c3*ZMAT^0*SIGMAT^2+
c4*ZMAT^1*SIGMAT^0+c5*ZMAT^1*SIGMAT^1+c6*ZMAT^1*SIGMAT^2+
c7*ZMAT^2*SIGMAT^0+c8*ZMAT^2*SIGMAT^1+c9*ZMAT^2*SIGMAT^2+
c10*s*ZMAT^0*SIGMAT^0+c11*s*ZMAT^0*SIGMAT^1+c12*s*ZMAT^0*SIGMAT^2+
c13*s*ZMAT^1*SIGMAT^0+c14*s*ZMAT^1*SIGMAT^1+c15*s*ZMAT^1*SIGMAT^2+
c16*s*ZMAT^2*SIGMAT^0+c17*s*ZMAT^2*SIGMAT^1+c18*s*ZMAT^2*SIGMAT^2$

% Explicitly:

AA:=(1/20)*mat(
(2*(2*s*z^2+s*z-s-z^2-7*z),-3*s*z^2-3*s*z-4*s+2*z^2+8*z-2,2*(s*z^2+s*z-2*s+2*z^2+2)),
(12*s*(-z^2-z-3),2*(-2*s*z^2-5*s*z-7*s+3*z^2+5*z+8),2*(s*z^2+3*s*z+2*s-2*z^2-2*z-6)),
(3*(s*z^2-3*s*z+2*z^2-4*z-6),6*(s*z^2+s*z-2*s-2*z^2-2),4*(2*s*z-s-z^2+z+1)))$

BB:=(1/60)*mat(
(3*(-5*s*z^2-9*s*z-8*s+2*z^2+2),3*(3*s*z^2-3*s*z+2*s-2*z+6),-5*s*z^2-7*s*z-14*s-4*z^2-2*z+2),
(3*(5*s*z^2+5*s*z-2*z^2+4*z+6),3*(s*z^2+3*s*z+2*s-4*z^2+6*z-2),6*(-s*z^2-s+3*z^2+5*z+8)),
(9*(-s*z^2+3*s*z-6*z^2-8*z-2),6*(s*z-3*s+3*z^2-z+6),6*(2*s*z^2+3*s*z+3*s+z^2-3*z)))$

CC:=(1/2)*mat(
(z*(s*z+s-2),0,0),
(0,-s*z^2-2*s*z-3*s+z^2+z+2,0),
(0,0,s*z-s-z^2+z))$

AASTAR:=sub(s=-s,tp(AA))$
BBSTAR:=sub(s=-s,tp(BB))$
CCSTAR:=sub(s=-s,tp(CC))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;
CCSTAR*FF*CC-FF;

% Checking the integrality conditions:

condmatq2type1:=mat(
(1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0),
(0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0),
(0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0),
(0,0,3,0,0,6,0,0,6,0,0,0,0,0,6,0,0,0),
(1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0),
(0,1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0),
(0,3,0,0,6,0,0,6,0,0,0,0,0,2,0,0,0,0),
(0,0,3,0,0,6,0,0,6,0,0,0,0,0,2,0,0,0),
(1,0,0,2,0,0,2,0,0,0,0,0,6,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0),
(0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0),
(0,0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1),
(0,0,0,0,0,5,0,0,4,0,0,3,0,0,6,0,0,6),
(0,0,0,7,0,0,4,0,0,1,0,0,2,0,0,2,0,0),
(0,0,0,0,7,0,0,4,0,0,1,0,0,2,0,0,2,0),
(0,0,0,0,3,0,0,4,0,0,3,0,0,6,0,0,6,0),
(0,0,0,0,0,3,0,0,4,0,0,3,0,0,6,0,0,6),
(0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,0))$

condmatq5type1:=mat(
(1,12,19,0,19,3,16,11,7,0,0,0,0,8,4,0,13,19),
(0,0,0,1,19,3,6,14,18,0,0,0,0,4,2,0,24,12),
(0,0,0,0,0,0,5,20,15,0,0,0,0,5,15,0,10,5),
(0,19,3,16,11,7,11,0,0,0,8,4,0,13,19,0,21,23),
(1,19,3,6,14,18,17,23,1,0,4,2,0,24,12,0,18,9),
(0,0,0,5,20,15,20,5,10,0,5,15,0,10,5,0,5,15),
(0,11,7,13,21,2,22,23,1,0,3,14,0,23,24,0,2,1),
(0,0,0,18,17,4,20,5,10,0,10,5,0,2,1,0,20,10),
(1,19,3,18,17,4,14,16,17,0,21,23,0,18,9,0,19,22),
(0,0,0,0,21,23,0,6,3,1,12,19,0,19,3,16,11,7),
(0,0,0,0,23,24,0,13,19,0,0,0,1,19,3,6,14,18),
(0,0,0,0,10,5,0,20,10,0,0,0,0,0,0,5,20,15),
(0,21,23,0,6,3,0,2,1,0,19,3,16,11,7,11,0,0),
(0,23,24,0,13,19,0,16,8,1,19,3,6,14,18,17,23,1),
(0,10,5,0,20,10,0,10,5,0,0,0,5,20,15,20,5,10),
(0,11,18,0,1,13,0,24,12,0,11,7,13,21,2,22,23,1),
(0,20,10,0,24,12,0,15,20,0,0,0,18,17,4,20,5,10),
(0,2,1,0,16,8,0,3,14,1,19,3,18,17,4,14,16,17))$

avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18))$

bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$

cvec:=mat((c1),(c2),(c3),(c4),(c5),(c6),(c7),(c8),(c9),(c10),(c11),(c12),(c13),(c14),(c15),(c16),(c17),(c18))$

% The following are zero, checking that condmatq2type1*avec, condmatq2type1*bvec and condmatq2type1*cvec
% have entries in $\Z_2$:
condmatq2type1*avec -(1/5)*mat((-11),(6),(2),(12),(-7),(3),(12),(10),(-5),(2),(-4),(1),(7),(-24),(12),(-4),(7),(-3));
condmatq2type1*bvec -(1/15)*mat((3),(0),(-2),(-18),(-9),(-3),(0),(-11),(-36),(-30),(-3),(-10),(-35),(-21),(-9),(0),(-34),(-21));
condmatq2type1*cvec - mat((-3),(0),(0),(0),(-1),(0),(0),(0),(1),(2),(0),(0),(0),(-5),(0),(0),(0),(1));

% The following are zero, checking that condmatq5type1*avec, condmatq5type1*bvec and condmatq5type1*cvec
% have entries in $\Z_5$:
condmatq5type1*avec - (1/4)*mat((41),(34),(17),(-50),(-18),(3),(-19),(-42),(-59),(12),(51),(30),(15),(27),(53),(14),(76),(27));
condmatq5type1*bvec - (1/12)*mat((26),(-10),(-23),(30),(29),(-81),(-66),(13),(-20),(-154),(-107),(-16),(-59),(-21),(-139),(-112),(-168),(-180));
condmatq5type1*cvec - mat((0),(-1),(0),(-16),(-6),(-5),(-13),(-18),(-18),(8),(7/2),(5/2),(27/2),(23/2),(25/2),(35/2),(19),(16));

% Checking the determinants. The following are zero:

det(AA)-1;
det(BB)-(1+2*s)/3;
det(CC)-1;


% Checking the relations:

% Here are the inverses of AA, BB and CC:

AAI:=mat(
((-2*s*z^2-s*z+s-z^2-7*z)/10,(-s*z^2+3*s*z+2*z^2-4*z-6)/20,(s*(z^2+z+3))/5),
((3*(-s*z^2-s*z+2*s+2*z^2+2))/10,(-2*s*z+s-z^2+z+1)/5,(-s*z^2-3*s*z-2*s-2*z^2-2*z-6)/10),
((3*(3*s*z^2+3*s*z+4*s+2*z^2+8*z-2))/20,(3*(-s*z^2-s*z+2*s-2*z^2-2))/10,(2*s*z^2+5*s*z+7*s+3*z^2+5*z+8)/10))$

BBI:=mat(
((5*s*z^2+9*s*z+8*s+2*z^2+2)/20,(s*z^2-3*s*z-6*z^2-8*z-2)/20,(-5*s*z^2-5*s*z-2*z^2+4*z+6)/60),
((5*s*z^2+7*s*z+14*s-4*z^2-2*z+2)/20,(-2*s*z^2-3*s*z-3*s+z^2-3*z)/10,(s*z^2+s+3*z^2+5*z+8)/10),
((3*(-3*s*z^2+3*s*z-2*s-2*z+6))/20,(-s*z+3*s+3*z^2-z+6)/10,(-s*z^2-3*s*z-2*s-4*z^2+6*z-2)/20))$

CCI:=mat(((z*(-s*z-s-2))/2,0,0),
         (0,(-s*z+s-z^2+z)/2,0),
         (0,0,(s*z^2+2*s*z+3*s+z^2+z+2)/2))$

% The following matrices are zero:

BB^3 - gendt*ID3;
(BB*CC*AAI*CC*AA)^3 - gendt*ID3;
AAI*BBI*AAI*CC*BB*CC*AA*BB*CC*AAI*CC*BBI - ID3;
AAI*BBI*AAI*CC*AAI*BB*CC*AAI*BBI*AAI^2*BB - ID3;
CCI*AAI*BBI*CCI*AAI*BBI*CCI*BBI*AA*BB*CC*AAI*BBI - gendti*ID3;
BB*CC*AAI*BBI*CCI^2*BBI*CCI*AA*CCI*BBI*AA*CCI*BBI*AAI - gendti*ID3;
AA*BB*AA*BB*CC*AAI*CC*BBI*CC*AAI*CC*BB*CC*BB*AA^2*CCI - gendt*ID3;
AA^2*CCI*AA*CCI*BBI*AA*BB*CC^2*AAI^2*BBI*AA*BB*CC*AAI*CC - ID3;
AA*CCI*AA*CCI*BBI*AA*BB*CC*BBI*AAI*CC*AAI^2*BBI*CC*AAI^2*BBI*CCI - gendti*ID3;
BB*AA^2*CCI*AA*BB*AA*BBI*CCI*AA*CCI*BBI*AAI*CC*AAI*CC*BBI*CC*BB*AA - ID3;
AAI*BBI*CCI*BBI*AA^2*BB*CC*AAI^2*BBI*AAI*BBI*AAI*CC*BBI*CC*AAI*CC*BB*CC - gendti*ID3;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  The generators for $\bar\Gamma$ in the case  $(a=2,p=3,\{2\})$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

ad1:=0/20$
ad2:=-2/20$
ad3:=4/20$
ad4:=-14/20$
ad5:=8/20$
ad6:=0/20$
ad7:=-2/20$
ad8:=2/20$
ad9:=4/20$
ad10:=-2/20$
ad11:=-4/20$
ad12:=-4/20$
ad13:=2/20$
ad14:=-3/20$
ad15:=2/20$
ad16:=4/20$
ad17:=-3/20$
ad18:=2/20$

bd1:=0/60$
bd2:=18/60$
bd3:=12/60$
bd4:=6/60$
bd5:=-15/60$
bd6:=2/60$
bd7:=18/60$
bd8:=3/60$
bd9:=-2/60$
bd10:=18/60$
bd11:=9/60$
bd12:=18/60$
bd13:=-6/60$
bd14:=-3/60$
bd15:=10/60$
bd16:=0/60$
bd17:=0/60$
bd18:=8/60$

cd1:=0/2$
cd2:=0/2$
cd3:=0/2$
cd4:=-2/2$
cd5:=0/2$
cd6:=0/2$
cd7:=0/2$
cd8:=0/2$
cd9:=0/2$
cd10:=0/2$
cd11:=0/2$
cd12:=0/2$
cd13:=1/2$
cd14:=0/2$
cd15:=0/2$
cd16:=1/2$
cd17:=0/2$
cd18:=0/2$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

AD:=ad1*ZMAT^0*SIGMAT^0+ad2*ZMAT^0*SIGMAT^1+ad3*ZMAT^0*SIGMAT^2+
ad4*ZMAT^1*SIGMAT^0+ad5*ZMAT^1*SIGMAT^1+ad6*ZMAT^1*SIGMAT^2+
ad7*ZMAT^2*SIGMAT^0+ad8*ZMAT^2*SIGMAT^1+ad9*ZMAT^2*SIGMAT^2+
ad10*s*ZMAT^0*SIGMAT^0+ad11*s*ZMAT^0*SIGMAT^1+ad12*s*ZMAT^0*SIGMAT^2+
ad13*s*ZMAT^1*SIGMAT^0+ad14*s*ZMAT^1*SIGMAT^1+ad15*s*ZMAT^1*SIGMAT^2+
ad16*s*ZMAT^2*SIGMAT^0+ad17*s*ZMAT^2*SIGMAT^1+ad18*s*ZMAT^2*SIGMAT^2$

BD:=bd1*ZMAT^0*SIGMAT^0+bd2*ZMAT^0*SIGMAT^1+bd3*ZMAT^0*SIGMAT^2+
bd4*ZMAT^1*SIGMAT^0+bd5*ZMAT^1*SIGMAT^1+bd6*ZMAT^1*SIGMAT^2+
bd7*ZMAT^2*SIGMAT^0+bd8*ZMAT^2*SIGMAT^1+bd9*ZMAT^2*SIGMAT^2+
bd10*s*ZMAT^0*SIGMAT^0+bd11*s*ZMAT^0*SIGMAT^1+bd12*s*ZMAT^0*SIGMAT^2+
bd13*s*ZMAT^1*SIGMAT^0+bd14*s*ZMAT^1*SIGMAT^1+bd15*s*ZMAT^1*SIGMAT^2+
bd16*s*ZMAT^2*SIGMAT^0+bd17*s*ZMAT^2*SIGMAT^1+bd18*s*ZMAT^2*SIGMAT^2$

CD:=cd1*ZMAT^0*SIGMAT^0+cd2*ZMAT^0*SIGMAT^1+cd3*ZMAT^0*SIGMAT^2+
cd4*ZMAT^1*SIGMAT^0+cd5*ZMAT^1*SIGMAT^1+cd6*ZMAT^1*SIGMAT^2+
cd7*ZMAT^2*SIGMAT^0+cd8*ZMAT^2*SIGMAT^1+cd9*ZMAT^2*SIGMAT^2+
cd10*s*ZMAT^0*SIGMAT^0+cd11*s*ZMAT^0*SIGMAT^1+cd12*s*ZMAT^0*SIGMAT^2+
cd13*s*ZMAT^1*SIGMAT^0+cd14*s*ZMAT^1*SIGMAT^1+cd15*s*ZMAT^1*SIGMAT^2+
cd16*s*ZMAT^2*SIGMAT^0+cd17*s*ZMAT^2*SIGMAT^1+cd18*s*ZMAT^2*SIGMAT^2$

ADSTAR:=sub(s=-s,tp(AD))$
BDSTAR:=sub(s=-s,tp(BD))$
CDSTAR:=sub(s=-s,tp(CD))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
ADSTAR*FF*AD-FF;
BDSTAR*FF*BD-FF;
CDSTAR*FF*CD-FF;

% Checking the integrality conditions:

condmatq2type2:=mat(
(1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,0),
(0,1,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,2,0,0,6,0,0,2),
(0,0,3,0,0,6,0,0,6,0,0,0,0,0,6,0,0,0),
(1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0),
(0,0,0,0,6,0,0,0,0,0,2,0,0,4,0,0,4,0),
(0,0,0,0,5,0,0,4,0,0,5,0,0,2,0,0,2,0),
(0,0,0,0,0,5,0,0,4,0,0,5,0,0,2,0,0,2),
(1,0,0,2,0,0,2,0,0,0,0,0,6,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0),
(0,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0),
(0,0,7,0,0,5,0,0,7,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,5,0,0,4,0,0,3,0,0,6,0,0,6),
(0,0,0,7,0,0,4,0,0,1,0,0,2,0,0,2,0,0),
(0,7,0,0,6,0,0,6,0,0,0,0,0,6,0,0,0,0),
(0,3/2,0,0,7,0,0,3,0,0,0,0,0,5,0,0,4,0),
(0,0,3/2,0,0,7,0,0,3,0,0,0,0,0,5,0,0,4),
(0,0,0,1,0,0,4,0,0,1,0,0,2,0,0,2,0,0))$

advec:=mat((ad1),(ad2),(ad3),(ad4),(ad5),(ad6),(ad7),(ad8),(ad9),(ad10),(ad11),(ad12),(ad13),(ad14),(ad15),(ad16),(ad17),(ad18))$

bdvec:=mat((bd1),(bd2),(bd3),(bd4),(bd5),(bd6),(bd7),(bd8),(bd9),(bd10),(bd11),(bd12),(bd13),(bd14),(bd15),(bd16),(bd17),(bd18))$

cdvec:=mat((cd1),(cd2),(cd3),(cd4),(cd5),(cd6),(cd7),(cd8),(cd9),(cd10),(cd11),(cd12),(cd13),(cd14),(cd15),(cd16),(cd17),(cd18))$

% The following are zero, checking that condmatq2type2*advec, condmatq2type2*bdvec and condmatq2type2*cdvec
% have entries in $\Z_2$:
condmatq2type2*advec - (1/5)*mat((-11),(6),(2),(12),(-7),(4),(4),(1),(-5),(2),(-4),(14),(7),(-24),(7),(8),(9),(-3));
condmatq2type2*bdvec - (1/15)*mat((9),(-6),(28),(24),(9),(-21),(-6),(32),(3),(0),(0),(20),(41),(30),(9),(-21),(27),(21));
condmatq2type2*cdvec - mat((-3),(0),(0),(0),(-1),(0),(0),(0),(1),(2),(0),(0),(0),(-5),(0),(0),(0),(1));

% The following are zero, checking that condmatq5type1*advec, condmatq5type1*bdvec and condmatq5type1*cdvec
% have entries in $\Z_5$:
condmatq5type1*advec - (1/4)*mat((41),(34),(17),(-50),(-18),(3),(-19),(-42),(-59),(12),(51),(30),(15),(27),(53),(14),(76),(27));
condmatq5type1*bdvec -(1/12)*mat((128),(-11),(59),(200),(161),(114),(194),(96),(264),(48),(-37),(4),(148),(186),(24),(117),(22),(13));
condmatq5type1*cdvec - mat((0),(-1),(0),(-16),(-6),(-5),(-13),(-18),(-18),(8),(7/2),(5/2),(27/2),(23/2),(25/2),(35/2),(19),(16));

% Checking the determinants. The following are zero:

det(AD)-1;
det(BD)-(1+2*s)/3;
det(CD)-1;

% Here are the inverses of AD, BD and CD:

ADI:=mat(
((-2*s*z^2-s*z+s-z^2-7*z)/10,(-s*z^2+3*s*z+2*z^2-4*z-6)/20,(s*(z^2+z+3))/5),
((3*(-s*z^2-s*z+2*s+2*z^2+2))/10,(-2*s*z+s-z^2+z+1)/5,(-s*z^2-3*s*z-2*s-2*z^2-2*z-6)/10),
((3*(3*s*z^2+3*s*z+4*s+2*z^2+8*z-2))/20,(3*(-s*z^2-s*z+2*s-2*z^2-2))/10,(2*s*z^2+5*s*z+7*s+3*z^2+5*z+8)/10))$

BDI:=mat(
((s*z-3*s+3*z^2+z)/10,(-2*s*z^2-5*s*z-7*s-z^2+z+6)/20,(s*z^2+s*z-2*s+4*z^2-2*z+10)/30),
((-4*s*z^2-5*s*z-9*s-z^2+z+6)/10,(-s*z^2-s*z-3*s-z^2-z-3)/5,(s*z^2+2*s*z+2*z+4)/10),
((3*(s*z-3*s+z^2-5*z+6))/20,(3*s*z^2+4*s*z+s-3*z^2+z+4)/10,(2*s*z^2+s*z-s-z^2+z-4)/10))$

CDI:=mat(((z*(-s*z-s-2))/2,0,0),
         (0,(-s*z+s-z^2+z)/2,0),
         (0,0,(s*z^2+2*s*z+3*s+z^2+z+2)/2))$

(BD*AD)^3 - gendt*ID3;
(CDI*BDI*CDI*AD)^3 - gendti*ID3;
(AD*BD*CD*ADI*CD*AD)^3 - gendt*ID3;
CD*ADI^2*BDI*CDI*BD*CD*ADI*BD*CDI*BDI*ADI - ID3;
BD*CD*ADI*CD*AD*BD*CD*ADI*CD*BDI*CDI*BDI*ADI*CD - ID3;
AD*CDI*BDI*CD*BD^2*CD*BD*CD*AD*CDI*BDI^2*CDI - ID3;
ADI*BDI*CDI*BDI*CDI*BDI^2*CDI*AD^2*BD*CD*BD*AD*CD*BD^2*CD - ID3;
BDI*AD*CDI*BDI^2*CDI*BDI*CDI*AD*CDI*AD*BD*AD*CDI*BDI*CDI*BDI^2 - gendti^2*ID3;
BD*ADI^2*BDI*CDI*BDI*CDI*BDI^2*CD*ADI^2*CD*BD^2*CD*BD*CD - ID3;
BDI^2*CDI*BD*CD*ADI^2*CD*BDI*AD*CDI*BDI^2*CDI*BDI^2*AD*CDI - gendti^2*ID3;
BD^3*CD*BD*CD*BD*AD^3*BD*CD*ADI*CD*BDI*AD*CDI*BDI^2*CDI - gendt*ID3;
ADI*BDI*CDI*BDI*CDI*BDI*CD*BD*CD*BD*AD*CD*BD*CD*BD^2*CD*ADI*BD*CDI - gendt*ID3;
BDI*CD*BD*CD*BD*CDI*ADI*CD*BD*CDI*AD*CDI*BDI*AD*CDI*BD*CDI*AD*CDI*BDI*ADI*BDI*AD*CDI - ID3;


% Finding generators of the intersection of the two groups.
% Note that the following are zero:

AA-AD;
CC-CD;
BBI*CC*BB*AA-BDI*CDI*BD*CD*ADI;
BBI*AAI*BBI-BDI*CDI*BDI;

;end;