Integral Representations of Z_23.sagews

def degree(elt, ring=None):
    if ring is None:
        return elt.polynomial().degree()
    if ring is not None:
        try:
            elt = elt.base_extend(ring)
        except TypeError as err:
            print(err)
            print("Define a coercion homomorphism from " + str(elt.parent()) + " to " + str(ring))
        else:
            return elt.polynomial().degree()

K.<t> = CyclotomicField(23)

assert degree(t^13 + 11*t) == 13, "13"
assert degree(3/4*t^6 + 12*t^8+3*t * t^9) == 10, "10"
assert degree(K(1/3)) == 0, "0"
degree(sqrt(2), ring=K)

no base extension defined Define a coercion homomorphism from Symbolic Ring to Cyclotomic Field of order 23 and degree 22

def coefficient(elt,basis_vector):
    higher_powers = elt.list()[degree(basis_vector)+1:]
    if sum([abs(x) for x in higher_powers]):
        raise Exception("Possible non-unique decomposition! "+ str(higher_powers))
    if elt.list()[degree(basis_vector)] == 0:
        return 0
    coeff_elt = elt.list()[degree(basis_vector)]
    highest_coeff_basis_vect = basis_vector.list()[degree(basis_vector)]
    coeff = coeff_elt/highest_coeff_basis_vect
    return coeff

K.<t> = CyclotomicField(5)
basis = [t^2 + 1, t^4 + 2, t^3 + t + 2]
elt = basis[0] + 3/4*basis[1]
assert coefficient(elt, basis[1]) == 3/4, "3/4"
elt = elt - coefficient(elt, basis[1])*basis[1]
assert coefficient(elt, basis[0]) == 1, "1"
assert coefficient(elt, basis[2]) == 0, "0"

def decompose(elt, basis, integer=True):
    decomposition_vector = [0 for _ in basis]

    while elt != 0:
        for index, basis_elt in enumerate(basis):
            if degree(basis_elt) == degree(elt):
                break
        else:
            raise Exception("Cannot decompose " + str(elt)  + ": no polynomial of matching degree in basis!\n" + str(basis))

        coeff = coefficient(elt, basis_elt)
        if integer:
            assert coeff.is_integer(), str(elt) + " is NOT decomposable over integers!"
        assert decomposition_vector[index] == 0, "Non-unique decomposition!"
        decomposition_vector[index] = coeff
        # print(elt, coeff,basis_elt)
        elt -= coeff*basis_elt
        # print(decomposition_vector)

    return decomposition_vector

K.<t> = NumberField(cyclotomic_polynomial(23))
basis = [t^5 + 2, 13*t -5, 6*t^2+ 2*t + 3]
elt_coords = [-2,3,1]
dot = lambda a,b : sum([x[0]*x[1] for x in zip(a,b)])
elt = dot(basis,elt_coords)
assert decompose(elt, basis) == elt_coords, str(basis)

assert decompose(t^13+11*t, K.ideal((47, t + 20)).integral_basis()) == [-5, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], "t^13 + 11*t"

try:
    decompose(1/3*t^5 + 5*t^8, K.ideal((47, t + 20)).integral_basis())
except AssertionError as err:
    print(err)
try:
    decompose(3*basis[0] -2*basis[2] - t^4, basis)
except Exception as exc:
    print(exc)

decompose(3*basis[0] -2*basis[2] + basis[1], basis)

1/3*t^5 - 220 is NOT decomposable over integers! Cannot decompose -t^4 - 12*t^2 - 4*t - 6: no polynomial of matching degree in basis! [t^5 + 2, 13*t - 5, 6*t^2 + 2*t + 3] [3, 1, -2]

def matrix_from_ideal(ideal):
    basis = ideal.integral_basis()
    t = ideal.ring().gen()
    shifted_basis = [t*v for v in basis]
    m = matrix([decompose(elt, basis) for elt in shifted_basis])
    return m

K.<t> = CyclotomicField(11)
I = K.ideal(1)
I.gens()
matrix_from_ideal(I)

(1,) [ 0 1 0 0 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0] [ 0 0 0 1 0 0 0 0 0 0] [ 0 0 0 0 1 0 0 0 0 0] [ 0 0 0 0 0 1 0 0 0 0] [ 0 0 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 0 0 1] [-1 -1 -1 -1 -1 -1 -1 -1 -1 -1]

show(cyclotomic_polynomial(23))
K.<t> = CyclotomicField(23)
K_IdealClassGrp = K.class_group(proof=False);

## takes ~ 1.5h to compute with proof=True (default)
# K_IdealClassGrp = K.class_group();

I = K_IdealClassGrp.gen();
show(I.gens())

m = matrix_from_ideal((I^1).ideal())
assert m^23 == matrix.identity(22)
assert [QQ(round(x^23,6)) for x in m.eigenvalues()] == 22*[1]

$\displaystyle x^{22} + x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1$

($\displaystyle 47$, $\displaystyle t + 30$, $\displaystyle t^{2} + 40$, $\displaystyle t^{3} + 22$, $\displaystyle t^{4} + 45$, $\displaystyle t^{5} + 13$, $\displaystyle t^{6} + 33$, $\displaystyle t^{7} + 44$, $\displaystyle t^{8} + 43$, $\displaystyle t^{9} + 26$, $\displaystyle t^{10} + 19$, $\displaystyle t^{11} + 41$, $\displaystyle t^{12} + 39$, $\displaystyle t^{13} + 5$, $\displaystyle t^{14} + 38$, $\displaystyle t^{15} + 35$, $\displaystyle t^{16} + 31$, $\displaystyle t^{17} + 10$, $\displaystyle t^{18} + 29$, $\displaystyle t^{19} + 23$, $\displaystyle t^{20} + 15$, $\displaystyle t^{21} + 20$)

(I^1).ideal().integral_basis()
(I^2).ideal().integral_basis()
(I^3).ideal().integral_basis()

[47, t + 30, t^2 + 40, t^3 + 22, t^4 + 45, t^5 + 13, t^6 + 33, t^7 + 44, t^8 + 43, t^9 + 26, t^10 + 19, t^11 + 41, t^12 + 39, t^13 + 5, t^14 + 38, t^15 + 35, t^16 + 31, t^17 + 10, t^18 + 29, t^19 + 23, t^20 + 15, t^21 + 20] [47, t + 44, t^2 + 38, t^3 + 20, t^4 + 13, t^5 + 39, t^6 + 23, t^7 + 22, t^8 + 19, t^9 + 10, t^10 + 30, t^11 + 43, t^12 + 35, t^13 + 11, t^14 + 33, t^15 + 5, t^16 + 15, t^17 + 45, t^18 + 41, t^19 + 29, t^20 + 40, t^21 + 26] [1, t, t^2, t^3, t^4, t^5, t^6, t^7, t^8, t^9, t^10, t^11, t^12, t^13, t^14, t^15, t^16, t^17, t^18, t^19, t^20, t^21]

show(m)

$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrr} -30 & 47 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -20 & 30 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -26 & 40 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -15 & 22 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -29 & 45 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -9 & 13 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -22 & 33 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -29 & 44 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -28 & 43 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -17 & 26 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -13 & 19 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -27 & 41 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -25 & 39 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -4 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -25 & 38 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -23 & 35 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ -20 & 31 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ -7 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ -19 & 29 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ -15 & 23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -10 & 15 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 19 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \end{array}\right)$

L.<t> = CyclotomicField(29)
L_class_g = L.class_group(proof=False);
L_class_g
L_class_g.gens()

Class group of order 8 with structure C2 x C2 x C2 of Cyclotomic Field of order 29 and degree 28 (Fractional ideal class (59, t - 26), Fractional ideal class (59, t - 20), Fractional ideal class (59, t - 7))

m = matrix_from_ideal(L_class_g.1.ideal())

show(m)

$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrrrrrr} -39 & 59 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -26 & 39 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -9 & 13 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -16 & 24 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -6 & 8 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -28 & 42 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -10 & 14 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -30 & 44 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -36 & 54 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -12 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -4 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -27 & 40 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -22 & 33 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -8 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -29 & 43 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -23 & 34 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -21 & 31 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -20 & 30 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -7 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -16 & 23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -32 & 47 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ -37 & 55 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ -26 & 38 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ -35 & 52 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ -25 & 37 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -22 & 32 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ -19 & 49 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \end{array}\right)$