# SageMath code for working with number field 28.28.819569564076950716311987772907236898045117333504.1

# (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 - 29*x^26 + 377*x^24 - 2900*x^22 + 14674*x^20 - 51359*x^18 + 127281*x^16 - 224808*x^14 + 281010*x^12 - 243542*x^10 + 140998*x^8 - 51272*x^6 + 10556*x^4 - 1015*x^2 + 29)

# 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)]