Find nth_root on modulo p

cube_root.sage

def cube_root(c, q):
    F = FiniteField(q)
    R.<x> = PolynomialRing(F,'x')
    while 1:
        a = F.random_element()
        b = F.random_element()
        fx = x**3 - a*x**2 + b*x - c
        fc = list(factor(fx))
        if len(fc) <= 1:
            root = pow(x, (q**2+q+1)//3, fx)
            root %= x
            return int(root)

def cube_roots(c,mod):
    c = c % mod
    rems = []
    if gcd( (mod-1), 3) == 1:
        d = inverse_mod(3, mod - 1)
        rems.append( int(pow(c, d, mod)) )
    else:
        g = GF(mod).multiplicative_generator()
        u = int(g ** ((mod-1)//3))
        r1 = int(cube_root(c, mod))
        for i in range(3):
            rems.append( int(r1 * pow(u, i, mod) % mod) )
    for m in rems :
        if pow(m,3,p) != c :
            print('%d = m^3 mod %d, m has no integer solutions.' % (c,p) )
            exit()
    return rems


if __name__ == '__main__':

    # Debug
    # p = random_prime(2^128-1,False,2^127)
    # x = randint(2^127,2^128) % p
    # c = pow(x, 3, p)
    # print(f"{x}^{3} mod {p} = {c}")
    # rems = cube_roots(c%p, p)
    # print(f'x = {rems}')
    # assert x in rems

    print("x^3 mod p = c")
    c = int(input("c = "))
    p = int(input("p = "))
    rems = cube_roots(c%p,p)
    print(f"x = {rems}")

rth_root.sage

def get_phi(N):
    phi = N
    for f in factor(N):
        phi = phi * (1 - 1 / f[0])
    return phi

def get_roots(r,c,mod):
    rems = []
    if gcd( get_phi(mod), r) == 1:
        d = inverse_mod( r,get_phi(mod) )
        rems.append(int(pow(c, d, mod)))
    else:
        g = GF(mod).multiplicative_generator()
        u = int(g ** ((mod-1)/r))
        r1 = int(rth_root(r,c, mod))
        for i in range(r):
            rems.append( int(r1 * pow(u, i, mod) % mod) )
    return rems

def rth_root(c,p,root):
    rems = get_roots(root, c%p, p)
    for m in rems :
        if pow(m,root,p) != c :
            print('%d = m^%d mod %d, m has no integer solutions.' % (c,root,p) )
            exit()
    return rems
    
if __name__ == "__main__":
  
    # # Debug
    # p = random_prime(2^128-1,False,2^127)
    # x = randint(2^127,2^128) % p
    # e = random_prime(15)
    # c = pow(x,e,p)
    # print(f"{x}^{e} mod {p} = {c}")
    # rems = rth_root(c,p,e)
    # print(f'x = {rems}')
    # assert x in rems

    # Example 
    print("x^e mod p = c")
    c = int(input("c = "))
    p = int(input("p = "))
    e = int(input("e = "))
    rems = rth_root(c,p,e)
    print(f"x = {rems}")

square_root.sage

z = Mod(c, n)
print(sqrt(z))