test.sagews

K.<sqrt3>=NumberField(x^2-3)

def g(a,b,c,d):
    g=matrix([[a+sqrt3*b,-d-sqrt3*c],[d-sqrt3*c,a-sqrt3*b]])
    return g

x = [g(a,b,c,d) for a in range(-6,6) for b in range(-6,(6)) for c in range(-6,(6)) for d in range(-6,(6)) if a**2+d**2-3*b**2-3*c**2==1]


S.<t,u,v,y>=QQ[]

P = g(t,u,v,y)

INEQS=[(g.transpose()*P*g-P).trace() for g in x]

def coeff(F):
    C=[]
    C.append(int(F.coefficient({t:0,u:0,v:0,y:0})))
    C.append(int(F.coefficient({t:1,u:0,v:0,y:0})))
    C.append(int(F.coefficient({t:0,u:1,v:0,y:0})))
    C.append(int(F.coefficient({t:0,u:0,v:1,y:0})))
    C.append(int(F.coefficient({t:0,u:0,v:0,y:1})))
    return C

ineqs=[]

for F in INEQS:
    ineqs.append(coeff(F))

poly=Polyhedron(ieqs=ineqs)

poly.Hrepresentation()
print("\n")
poly.Vrepresentation()

(An inequality (1, -2, 0, 0) x + 0 >= 0, An inequality (1, 0, -2, 0) x + 0 >= 0, An inequality (1, 0, 2, 0) x + 0 >= 0, An inequality (1, 2, 0, 0) x + 0 >= 0, An inequality (2, -3, -2, 0) x + 0 >= 0, An inequality (2, -3, 2, 0) x + 0 >= 0, An inequality (2, -2, -3, 0) x + 0 >= 0, An inequality (2, -2, 3, 0) x + 0 >= 0, An inequality (2, 2, -3, 0) x + 0 >= 0, An inequality (2, 2, 3, 0) x + 0 >= 0, An inequality (2, 3, -2, 0) x + 0 >= 0, An inequality (2, 3, 2, 0) x + 0 >= 0) (A line in the direction (0, 0, 0, 1), A vertex at (0, 0, 0, 0), A ray in the direction (4, -2, -1, 0), A ray in the direction (4, -1, -2, 0), A ray in the direction (4, 1, 2, 0), A ray in the direction (4, 2, 1, 0), A ray in the direction (5, -2, -2, 0), A ray in the direction (4, 2, -1, 0), A ray in the direction (5, 2, 2, 0), A ray in the direction (4, 1, -2, 0), A ray in the direction (4, -1, 2, 0), A ray in the direction (5, 2, -2, 0), A ray in the direction (5, -2, 2, 0), A ray in the direction (4, -2, 1, 0))