# SageMath code for working with number field 33.33.2250468870721257864915422491944078011918280366020799086767808384626842687481.1

# 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^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591)

# 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^33 - 8*x^32 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591)
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)]