# SageMath code for working with number field 32.0.9572290431813101812896740520288704359886188011817648342612467121075247600488097.1.
# 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 + 6*x^30 - 46*x^29 + 391*x^28 + 717*x^27 + 7428*x^26 + 20280*x^25 + 45840*x^24 - 961862*x^23 - 2081212*x^22 - 1399032*x^21 + 24756306*x^20 + 107460240*x^19 + 836109803*x^18 + 1524157119*x^17 - 267790372*x^16 - 7012926215*x^15 + 38560539181*x^14 + 50727599993*x^13 - 210387899697*x^12 - 610442701949*x^11 + 1209773620966*x^10 + 3197253545113*x^9 - 6473190002488*x^8 - 7464078960084*x^7 + 35665340406463*x^6 - 53074991558346*x^5 + 46767096306383*x^4 - 24811228492420*x^3 + 7674195186138*x^2 - 1554391051686*x + 272803795483)

# 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 + 6*x^30 - 46*x^29 + 391*x^28 + 717*x^27 + 7428*x^26 + 20280*x^25 + 45840*x^24 - 961862*x^23 - 2081212*x^22 - 1399032*x^21 + 24756306*x^20 + 107460240*x^19 + 836109803*x^18 + 1524157119*x^17 - 267790372*x^16 - 7012926215*x^15 + 38560539181*x^14 + 50727599993*x^13 - 210387899697*x^12 - 610442701949*x^11 + 1209773620966*x^10 + 3197253545113*x^9 - 6473190002488*x^8 - 7464078960084*x^7 + 35665340406463*x^6 - 53074991558346*x^5 + 46767096306383*x^4 - 24811228492420*x^3 + 7674195186138*x^2 - 1554391051686*x + 272803795483)
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 pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]