# SageMath code for working with number field 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1

# (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^42 + 42*x^40 + 819*x^38 + 9842*x^36 + 81585*x^34 + 494802*x^32 + 2272424*x^30 + 8069423*x^28 + 22428224*x^26 + 49085050*x^24 + 84669739*x^22 + 114704933*x^20 + 121049551*x^18 + 98190708*x^16 + 60019861*x^14 + 26865216*x^12 + 8444436*x^10 + 1749188*x^8 + 215453*x^6 + 13181*x^4 + 294*x^2 + 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)]