# SageMath code for working with number field 33.33.228343593450302703244344174036290254973199912242460469577320120681.1

# 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 - 48*x^31 + 488*x^30 + 889*x^29 - 13082*x^28 - 6681*x^27 + 203003*x^26 - 17780*x^25 - 2022483*x^24 + 832653*x^23 + 13572345*x^22 - 8326527*x^21 - 62657637*x^20 + 47704277*x^19 + 199663335*x^18 - 178099102*x^17 - 433344046*x^16 + 447058648*x^15 + 617907878*x^14 - 750473359*x^13 - 533823452*x^12 + 817138243*x^11 + 221279551*x^10 - 544192951*x^9 + 11597098*x^8 + 199867737*x^7 - 43176794*x^6 - 33036600*x^5 + 11713378*x^4 + 1505253*x^3 - 925381*x^2 + 72350*x + 1013)

# 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 - 48*x^31 + 488*x^30 + 889*x^29 - 13082*x^28 - 6681*x^27 + 203003*x^26 - 17780*x^25 - 2022483*x^24 + 832653*x^23 + 13572345*x^22 - 8326527*x^21 - 62657637*x^20 + 47704277*x^19 + 199663335*x^18 - 178099102*x^17 - 433344046*x^16 + 447058648*x^15 + 617907878*x^14 - 750473359*x^13 - 533823452*x^12 + 817138243*x^11 + 221279551*x^10 - 544192951*x^9 + 11597098*x^8 + 199867737*x^7 - 43176794*x^6 - 33036600*x^5 + 11713378*x^4 + 1505253*x^3 - 925381*x^2 + 72350*x + 1013)
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)]