class template:
    '''Template of a p-group cover'''
    def __init__(self, height, field, group, fcts, gp_action):
        self.height = height
        self.group = group
        self.gp_action = gp_action #action of the generators of the group on z[i]'s
        self.field = field
        n = height
        variable_names = ''
        for i in range(n):
            variable_names += 'z'+str(i)+','
        for i in range(n):
            variable_names += 'f'+str(i)
            if i!=n-1:
                variable_names += ','
        Rzf = PolynomialRing(field, 2*n, variable_names)
        z = Rzf.gens()[:n]
        f = Rzf.gens()[n:]
        self.fct_field = Rzf, z, f
        self.fcts = [Rzf(ff) for ff in fcts] #RHSs of the Artin-Schreier equations

def elementary_template(p, n):
    group = elementary_gp(p, n)
    field = GF(p)
    variable_names = ''
    for i in range(n):
        variable_names += 'z'+str(i)+','
    for i in range(n):
        variable_names += 'f'+str(i)
        if i!=n-1:
            variable_names += ','
    R = PolynomialRing(field, 2*n, variable_names)
    z = R.gens()[:n]
    f = R.gens()[n:]
    height = n
    fcts = [f[i] for i in range(n)]
    gp_action = [[z[j] + (i == j) for j in range(n)] for i in range(n)]
    return template(height, field, group, fcts, gp_action)

def heisenberg_template(p):
    group = heisenberg(p)
    field = GF(p)
    variable_names = ''
    n = 3
    for i in range(n):
        variable_names += 'z'+str(i)+','
    for i in range(n):
        variable_names += 'f'+str(i)
        if i!=n-1:
            variable_names += ','
    R = PolynomialRing(field, 2*n, variable_names)
    z = R.gens()[:n]
    f = R.gens()[n:]
    height = n
    fcts = [f[i] for i in range(n)]
    fcts[2] += (z[0] - z[1])*f[1]
    gp_action = [[z[0] + 1, z[1], z[2] + z[1]], [z[0] + 1, z[1] + 1, z[2]], [z[0], z[1], z[2] - 1]]
    return template(height, field, group, fcts, gp_action)