# SageMath code for working with number field 28.0.20444429365013571404907915184985572799247638801.1

# (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^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675)

# 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)]