# SageMath code for working with number field 28.0.650754790224967946962108515803180182241777.1

# (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 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229)

# 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)]