# SageMath code for working with number field 33.33.964748920938762847635574420140466077720720339834593232479190796888489.1

# 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 - 70*x^31 + 734*x^30 + 1439*x^29 - 27994*x^28 + 11289*x^27 + 571079*x^26 - 1005928*x^25 - 6614661*x^24 + 19440415*x^23 + 40980613*x^22 - 194665653*x^21 - 82258867*x^20 + 1125872365*x^19 - 536982871*x^18 - 3781576926*x^17 + 4155619240*x^16 + 6998281772*x^15 - 12201861484*x^14 - 5994133153*x^13 + 18812121252*x^12 + 75160599*x^11 - 16337602363*x^10 + 3902636199*x^9 + 8090155706*x^8 - 3031301627*x^7 - 2187750082*x^6 + 1012187926*x^5 + 277584302*x^4 - 153651191*x^3 - 8485173*x^2 + 8057502*x - 470213)

# 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 - 70*x^31 + 734*x^30 + 1439*x^29 - 27994*x^28 + 11289*x^27 + 571079*x^26 - 1005928*x^25 - 6614661*x^24 + 19440415*x^23 + 40980613*x^22 - 194665653*x^21 - 82258867*x^20 + 1125872365*x^19 - 536982871*x^18 - 3781576926*x^17 + 4155619240*x^16 + 6998281772*x^15 - 12201861484*x^14 - 5994133153*x^13 + 18812121252*x^12 + 75160599*x^11 - 16337602363*x^10 + 3902636199*x^9 + 8090155706*x^8 - 3031301627*x^7 - 2187750082*x^6 + 1012187926*x^5 + 277584302*x^4 - 153651191*x^3 - 8485173*x^2 + 8057502*x - 470213)
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)]