#y^2 = x^3 + x
#in Z_11 and Z_13

#a) Gruppen bestimmen
#b) ord von jedem Gruppenelement

MODULE = 13
A = 1

def invmod(a):
    for d in range(1, MODULE):
        r = (d * a) % MODULE
        if r == 1:
            break
    else:
        raise ValueError('%d has no inverse mod %d' % (a, p))
    return d

def calcGroupElements():
    res = []
    for y in range(MODULE):
        for x in range(MODULE):
            if pow(y, 2, MODULE) == (pow(x, 3, MODULE) + x) % MODULE:
                res.append([x, y])
    return res

def addPoint(point):
    x1 = point[0]
    y1 = point[1]

    s = (3*x1*x1+A)*invmod((2*y1)%MODULE)

    x2 = (pow(s,2)-2*x1) % MODULE
    y2 = ((s*(x1-x2))-y1) % MODULE

    return [int(x2),int(y2)]

def addPoints(p1,p2):

    s = (p2[1]-p1[1])*invmod((p2[0]-p1[0])%MODULE)

    x3 = (pow(s,2)-p1[0]-p2[0]) % MODULE
    y3 = (s*(p1[0]-x3)-p1[1]) % MODULE

    return [int(x3),int(y3)]

def calcOrderOfEl(point):
    if point == [0,0]:
        return 2

    stopPoint = [point[0],-point[1]%MODULE]
    curPoint = point
    i = 2

    while not(curPoint == stopPoint):
        if(point == curPoint):
            curPoint = addPoint(curPoint)
        else:
            curPoint = addPoints(curPoint,point)
        i+=1
    return i

groupElements = calcGroupElements()
print(groupElements, len(groupElements))

for point in groupElements:
    print("Order of",point,"\t",calcOrderOfEl(point))