def pgl2_a4_1(q, a):
'''
q is assumed to be 1 mod 6
'''
pgl2 = PGL(2, q)
field = pgl2.get_field()
identity = pgl2.identity()
unity_3_roots = [
x for x in field.get_all_elements()
if x * x + x + field.one() == field.zero()
]
zeta = unity_3_roots[0]
field_two = field.one() + field.one()
s2_1 = pgl2.create([[a, field.one()], [field_two * a * a, -a]])
if s2_1.det() == field.zero():
return []
s3 = pgl2.create([[zeta, field.zero()], [field.zero(), field.one()]])
s2_2 = s3 * s2_1 * s3.inverse()
s2_3 = s2_1 * s2_2
elements = [identity, s2_1, s2_2, s2_3]
elements_addition = [x * s3 for x in elements]
elements_addition += [x * s3 * s3 for x in elements]
elements += elements_addition
return elements
def pgl2_a4_2(q, a):
'''
q is assumed to be 5 mod 6
'''
pgl2 = PGL(2, q)
field = pgl2.get_field()
identity = pgl2.identity()
two = field.one() + field.one()
three = two + field.one()
nonsquare = SquareExtensionField.get_non_square_element(field)
nonsquare_3_roots = [
x for x in field.get_all_elements() if three * x * x == -nonsquare
]
x = nonsquare_3_roots[0]
bs = [
b for b in field.get_all_elements()
if b * b - two * b * x * x + three * three * x * x * x * x +
two * two * a * a * x * x == field.zero()
]
if len(bs) == 0:
return []
s2_1 = pgl2.create([[a, bs[0]], [field.one(), -a]])
s3 = pgl2.create([[x, -three * x * x], [field.one(), x]])
s2_2 = s3 * s2_1 * s3.inverse()
s2_3 = s2_1 * s2_2
elements = [identity, s2_1, s2_2, s2_3]
elements_addition = [x * s3 for x in elements]
elements_addition += [x * s3 * s3 for x in elements]
elements += elements_addition
return elements
def pgl2_a4_0(q, a):
'''
q is assumed to be 3 ** k
'''
pgl2 = PGL(2, q)
field = pgl2.get_field()
identity = pgl2.identity()
s2_1 = pgl2.create([[a, -a * a - field.one()], [field.one(), -a]])
s3 = pgl2.create([[field.one(), field.one()], [field.zero(), field.one()]])
s2_2 = s3 * s2_1 * s3.inverse()
s2_3 = s2_1 * s2_2
elements = [identity, s2_1, s2_2, s2_3]
elements_addition = [x * s3 for x in elements]
elements_addition += [x * s3 * s3 for x in elements]
elements += elements_addition
return elements