readi p,"Input the prime p";
w:=0;
F:=FiniteField(p);
for i in [2..p-1] do
a:=F!i;
S:={a};
for j in [2..p-1] do
  Include(~S,a^j);
end for;
if #S eq p-1 then
  w:=i;
  break;
end if;
end for;
print "p equals",p;
print "w equals",w;


Z:=Integers();
V2:=VectorSpace(F,2);
H22:=Hom(V2,V2);

//Case babb=baaa
mats1:={};

for y1 in [0..(p-1) div 2] do
for y2 in [0..p-1] do
for y3 in [0..p-1] do
for y4 in [0..p-1] do

new :=1;
index:=p^3*y1+p^2*y2+p*y3+y4;

A:=H22![y1,y2,y3,y4];

for a in [0..p-1] do
for b in [0..p-1] do

if a ne b and a ne p-b then

B:=H22![a,b,b,a];
C:=H22![a^4-b^4,2*a*b*(a^2-b^2),2*a*b*(a^2-b^2),a^4-b^4];
D:=B*A*C^-1;

z1:=Z!(D[1][1]);
z2:=Z!(D[1][2]);
z3:=Z!(D[2][1]);
z4:=Z!(D[2][2]);

ind1:=p^3*z1+p^2*z2+p*z3+z4;

if ind1 lt index then new:=0; end if;

B:=H22![a,b,-b,-a];
C:=H22![-a^4+b^4,-2*a*b*(a^2-b^2),2*a*b*(a^2-b^2),a^4-b^4];
D:=B*A*C^-1;

z1:=Z!(D[1][1]);
z2:=Z!(D[1][2]);
z3:=Z!(D[2][1]);
z4:=Z!(D[2][2]);

ind1:=p^3*z1+p^2*z2+p*z3+z4;

if ind1 lt index then new:=0; end if;

end if;

if new eq 0 then break; end if;
end for;
if new eq 0 then break; end if;
end for;

if new eq 1 then
  Include(~mats1,[y1,y2,y3,y4]);
end if;

end for;
end for;
if #mats1 eq (Gcd(p-1,3)*(p^2+3*p+11)+1)/2 then break; end if;
end for;
if #mats1 eq (Gcd(p-1,3)*(p^2+3*p+11)+1)/2 then break; end if;
end for;

//Case babb=wbaaa
mats2:={};

for y1 in [0..(p-1) div 2] do
for y2 in [0..p-1] do
for y3 in [0..p-1] do
for y4 in [0..p-1] do

new :=1;
index:=p^3*y1+p^2*y2+p*y3+y4;

A:=H22![y1,y2,y3,y4];

for a in [0..p-1] do
for b in [0..p-1] do

if a+b ne 0 then

B:=H22![a,b,w*b,a];
C:=H22![a^4-w^2*b^4,2*a*b*(a^2-w*b^2),2*w*a*b*(a^2-w*b^2),a^4-w^2*b^4];
D:=B*A*C^-1;

z1:=Z!(D[1][1]);
z2:=Z!(D[1][2]);
z3:=Z!(D[2][1]);
z4:=Z!(D[2][2]);

ind1:=p^3*z1+p^2*z2+p*z3+z4;

if ind1 lt index then new:=0; end if;


B:=H22![a,b,-w*b,-a];
C:=H22![-a^4+w^2*b^4,-2*a*b*(a^2-w*b^2),2*w*a*b*(a^2-w*b^2),a^4-w^2*b^4];
D:=B*A*C^-1;

z1:=Z!(D[1][1]);
z2:=Z!(D[1][2]);
z3:=Z!(D[2][1]);
z4:=Z!(D[2][2]);

ind1:=p^3*z1+p^2*z2+p*z3+z4;

if ind1 lt index then new:=0; end if;

end if;

if new eq 0 then break; end if;
end for;
if new eq 0 then break; end if;
end for;

if new eq 1 then
  Include(~mats2,[y1,y2,y3,y4]);
end if;

end for;
end for;
if #mats2 eq (Gcd(p-1,3)*(p^2+p+1)+5)/2 then break; end if;
end for;
if #mats2 eq (Gcd(p-1,3)*(p^2+p+1)+5)/2 then break; end if;
end for;

print #mats1,(Gcd(p-1,3)*(p^2+3*p+11)+1)/2;
print #mats2,(Gcd(p-1,3)*(p^2+p+1)+5)/2;

