# SageMath code for working with number field 21.5.26730926988131422081122304.1

# (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^21 - 5*x^20 + 11*x^19 - 16*x^18 + 25*x^17 - 38*x^16 + 36*x^15 - 25*x^14 + 18*x^13 - 4*x^12 - 3*x^11 - 17*x^10 - 23*x^9 - 12*x^8 + 55*x^7 + 18*x^6 - 46*x^5 - 17*x^4 + 19*x^3 + 9*x^2 - 2*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)]