# SageMath code for working with number field 32.32.71166659227026865952165739269552697151542080277732014965436610606901057.1

# 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 - 93*x^30 + 210*x^29 + 3538*x^28 - 12479*x^27 - 65768*x^26 + 350581*x^25 + 485064*x^24 - 5339623*x^23 + 2808135*x^22 + 44434017*x^21 - 84476060*x^20 - 170361963*x^19 + 681732194*x^18 - 77972047*x^17 - 2529499095*x^16 + 3133974961*x^15 + 3455889877*x^14 - 10578598738*x^13 + 3732760107*x^12 + 12702259726*x^11 - 15151938680*x^10 - 638400055*x^9 + 11610249674*x^8 - 6566502528*x^7 - 1290174274*x^6 + 2262714296*x^5 - 382096988*x^4 - 214409519*x^3 + 59339724*x^2 + 2830728*x + 14969)

# 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 - 93*x^30 + 210*x^29 + 3538*x^28 - 12479*x^27 - 65768*x^26 + 350581*x^25 + 485064*x^24 - 5339623*x^23 + 2808135*x^22 + 44434017*x^21 - 84476060*x^20 - 170361963*x^19 + 681732194*x^18 - 77972047*x^17 - 2529499095*x^16 + 3133974961*x^15 + 3455889877*x^14 - 10578598738*x^13 + 3732760107*x^12 + 12702259726*x^11 - 15151938680*x^10 - 638400055*x^9 + 11610249674*x^8 - 6566502528*x^7 - 1290174274*x^6 + 2262714296*x^5 - 382096988*x^4 - 214409519*x^3 + 59339724*x^2 + 2830728*x + 14969)
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)]