# SageMath code for working with number field 32.0.29079187190356527093230483395294879034427143331250176.1

# Some of these functions may take a long time to execute (this depends on the field).

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^32 - 4*x^31 + 42*x^30 - 134*x^29 + 754*x^28 - 1990*x^27 + 7802*x^26 - 17690*x^25 + 53094*x^24 - 108182*x^23 + 256762*x^22 - 492454*x^21 + 925356*x^20 - 1701130*x^19 + 2638782*x^18 - 4227686*x^17 + 6633341*x^16 - 6962910*x^15 + 13367372*x^14 - 10724860*x^13 + 11545802*x^12 - 23539896*x^11 + 3955128*x^10 - 7435992*x^9 + 19530996*x^8 + 2533056*x^7 - 5443776*x^6 - 1219904*x^5 + 651552*x^4 + 141056*x^3 - 11264*x^2 - 2816*x + 256)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Autmorphisms: 
K.automorphisms()

# 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()

# Analytic class number formula: 
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ);  K.<a> = NumberField(x^32 - 4*x^31 + 42*x^30 - 134*x^29 + 754*x^28 - 1990*x^27 + 7802*x^26 - 17690*x^25 + 53094*x^24 - 108182*x^23 + 256762*x^22 - 492454*x^21 + 925356*x^20 - 1701130*x^19 + 2638782*x^18 - 4227686*x^17 + 6633341*x^16 - 6962910*x^15 + 13367372*x^14 - 10724860*x^13 + 11545802*x^12 - 23539896*x^11 + 3955128*x^10 - 7435992*x^9 + 19530996*x^8 + 2533056*x^7 - 5443776*x^6 - 1219904*x^5 + 651552*x^4 + 141056*x^3 - 11264*x^2 - 2816*x + 256)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator();  RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))  

# Intermediate fields: 
K.subfields()[1:-1]

# Galois group: 
K.galois_group(type='pari')

# Frobenius cycle types: 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]