# SageMath code for working with number field 32.0.1033818800357475102568684143000990752577071087616.1

# (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^32 - 4*x^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 65536)

# 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)]