# SageMath code for working with number field 32.0.38897685648686306961584993323026037365043690280067733586177953.1

# 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 - x^31 + 2*x^30 + 60*x^29 - 53*x^28 + 99*x^27 + 1385*x^26 - 1071*x^25 + 1854*x^24 + 15802*x^23 - 10509*x^22 + 16583*x^21 + 95912*x^20 - 54402*x^19 + 75534*x^18 + 310800*x^17 - 136853*x^16 + 163941*x^15 + 541045*x^14 - 33297*x^13 + 98317*x^12 + 537573*x^11 + 431212*x^10 - 151090*x^9 + 496535*x^8 + 326343*x^7 + 132437*x^6 + 314657*x^5 + 175914*x^4 + 198989*x^3 - 20144*x^2 - 93555*x + 27583)

# 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 - x^31 + 2*x^30 + 60*x^29 - 53*x^28 + 99*x^27 + 1385*x^26 - 1071*x^25 + 1854*x^24 + 15802*x^23 - 10509*x^22 + 16583*x^21 + 95912*x^20 - 54402*x^19 + 75534*x^18 + 310800*x^17 - 136853*x^16 + 163941*x^15 + 541045*x^14 - 33297*x^13 + 98317*x^12 + 537573*x^11 + 431212*x^10 - 151090*x^9 + 496535*x^8 + 326343*x^7 + 132437*x^6 + 314657*x^5 + 175914*x^4 + 198989*x^3 - 20144*x^2 - 93555*x + 27583)
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)]