# SageMath code for working with number field 33.33.36584611296554742180833097810429342639777502523008874222975105176339833601.1

# 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 - x^32 - 96*x^31 + 217*x^30 + 3795*x^29 - 13403*x^28 - 74197*x^27 + 394231*x^26 + 599821*x^25 - 6350170*x^24 + 2832807*x^23 + 56803105*x^22 - 104532088*x^21 - 244229488*x^20 + 932758015*x^19 + 5618002*x^18 - 3890173018*x^17 + 4529747891*x^16 + 6495127532*x^15 - 18110944809*x^14 + 4574986912*x^13 + 26694143816*x^12 - 29645825157*x^11 - 6037403432*x^10 + 30417132332*x^9 - 15468969217*x^8 - 6737165737*x^7 + 8515927088*x^6 - 1017730658*x^5 - 1409177433*x^4 + 395068072*x^3 + 75602260*x^2 - 21210003*x - 2947097)

# 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 - x^32 - 96*x^31 + 217*x^30 + 3795*x^29 - 13403*x^28 - 74197*x^27 + 394231*x^26 + 599821*x^25 - 6350170*x^24 + 2832807*x^23 + 56803105*x^22 - 104532088*x^21 - 244229488*x^20 + 932758015*x^19 + 5618002*x^18 - 3890173018*x^17 + 4529747891*x^16 + 6495127532*x^15 - 18110944809*x^14 + 4574986912*x^13 + 26694143816*x^12 - 29645825157*x^11 - 6037403432*x^10 + 30417132332*x^9 - 15468969217*x^8 - 6737165737*x^7 + 8515927088*x^6 - 1017730658*x^5 - 1409177433*x^4 + 395068072*x^3 + 75602260*x^2 - 21210003*x - 2947097)
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)]