# Oscar code for working with number field 27.3.177801404718372423264819198352687104000000000000000000000000000000000000000000000000.6.
# If you have not already loaded the Oscar package, you should type "using Oscar;" before running the code below.
# Some of these functions may take a long time to execute (this depends on the field).


# Define the number field: 
Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200)

# Defining polynomial: 
defining_polynomial(K)

# Degree over Q: 
degree(K)

# Signature: 
signature(K)

# Discriminant: 
OK = ring_of_integers(K); discriminant(OK)

# Ramified primes: 
prime_divisors(discriminant((OK)))

# Autmorphisms: 
automorphisms(K)

# Integral basis: 
basis(OK)

# Class group: 
class_group(K)

# Unit group: 
UK, fUK = unit_group(OK)

# Unit rank: 
rank(UK)

# Generator for roots of unity: 
torsion_units_generator(OK)

# Fundamental units: 
[K(fUK(a)) for a in gens(UK)]

# Regulator: 
regulator(K)

# Analytic class number formula: 
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

# Intermediate fields: 
subfields(K)[2:end-1]

# Galois group: 
G, Gtx = galois_group(K); G, transitive_group_identification(G)

# Frobenius cycle types: 
# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]