# SageMath code for working with number field 27.27.33967771186402966366427549984621879030979398934601324527479889.7

# Some of these functions may take a long time to execute (this depends on the field).

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^27 - 198*x^25 - 69*x^24 + 16947*x^23 + 10215*x^22 - 822495*x^21 - 626499*x^20 + 24996195*x^19 + 20626740*x^18 - 496209987*x^17 - 393964668*x^16 + 6511386444*x^15 + 4364596737*x^14 - 55763583489*x^13 - 25774577970*x^12 + 298557947277*x^11 + 57580976835*x^10 - 910210926315*x^9 + 71942819904*x^8 + 1311914075382*x^7 - 284784790380*x^6 - 772175021373*x^5 + 205559479314*x^4 + 123582042393*x^3 - 20629945359*x^2 - 2138965308*x + 140428837)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Autmorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

# Class group: 
K.class_group().invariants()

# Unit group: 
UK = K.unit_group()

# Unit rank: 
UK.rank()

# Generator for roots of unity: 
UK.torsion_generator()

# Fundamental units: 
UK.fundamental_units()

# Regulator: 
K.regulator()

# Analytic class number formula: 
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ);  K.<a> = NumberField(x^27 - 198*x^25 - 69*x^24 + 16947*x^23 + 10215*x^22 - 822495*x^21 - 626499*x^20 + 24996195*x^19 + 20626740*x^18 - 496209987*x^17 - 393964668*x^16 + 6511386444*x^15 + 4364596737*x^14 - 55763583489*x^13 - 25774577970*x^12 + 298557947277*x^11 + 57580976835*x^10 - 910210926315*x^9 + 71942819904*x^8 + 1311914075382*x^7 - 284784790380*x^6 - 772175021373*x^5 + 205559479314*x^4 + 123582042393*x^3 - 20629945359*x^2 - 2138965308*x + 140428837)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator();  RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))  

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

# Galois group: 
K.galois_group(type='pari')

# Frobenius cycle types: 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]