# Oscar code for working with number field 27.27.2519933803975014472273213858760468555951745833259300216067364409.3.
# 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 - 333*x^25 - 333*x^24 + 35298*x^23 + 53946*x^22 - 1819068*x^21 - 3588075*x^20 + 52704576*x^19 + 127734434*x^18 - 907552539*x^17 - 2673287370*x^16 + 9278172984*x^15 + 34236633762*x^14 - 51653439189*x^13 - 270237162441*x^12 + 93428282529*x^11 + 1280114382555*x^10 + 527095739526*x^9 - 3356233487919*x^8 - 3330019648665*x^7 + 3845500135071*x^6 + 6646340649357*x^5 + 111023372826*x^4 - 4196847578151*x^3 - 2127671869662*x^2 - 113904515466*x + 74779074553)

# 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))

# Automorphisms: 
automorphism_group(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 = polynomial_ring(QQ); K, a = number_field(x^27 - 333*x^25 - 333*x^24 + 35298*x^23 + 53946*x^22 - 1819068*x^21 - 3588075*x^20 + 52704576*x^19 + 127734434*x^18 - 907552539*x^17 - 2673287370*x^16 + 9278172984*x^15 + 34236633762*x^14 - 51653439189*x^13 - 270237162441*x^12 + 93428282529*x^11 + 1280114382555*x^10 + 527095739526*x^9 - 3356233487919*x^8 - 3330019648665*x^7 + 3845500135071*x^6 + 6646340649357*x^5 + 111023372826*x^4 - 4196847578151*x^3 - 2127671869662*x^2 - 113904515466*x + 74779074553);
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);
degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

# 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]