# SageMath code for working with number field 27.1.78637606867438430727852801672920631138199.1

# (Note that not all these functions may be available, and some may take a long time to execute.)

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

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

# 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()

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

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