# SageMath code for working with number field 33.33.45897850273808078905473711375352471596627685094598222212495078792349891889.1

# 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 - 114*x^31 + 1061*x^30 + 5059*x^29 - 60653*x^28 - 96801*x^27 + 1959353*x^26 - 29225*x^25 - 39423204*x^24 + 40518048*x^23 + 513641811*x^22 - 918583926*x^21 - 4362858468*x^20 + 10893040871*x^19 + 23607826158*x^18 - 79952364670*x^17 - 75116228962*x^16 + 381639679930*x^15 + 98812353215*x^14 - 1205788951741*x^13 + 175885903114*x^12 + 2530475990857*x^11 - 987113141315*x^10 - 3498652499635*x^9 + 1841635007257*x^8 + 3110258033850*x^7 - 1787906886467*x^6 - 1685908305594*x^5 + 902138910730*x^4 + 495599994390*x^3 - 195854103442*x^2 - 57250484608*x + 8460460991)

# 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 - 114*x^31 + 1061*x^30 + 5059*x^29 - 60653*x^28 - 96801*x^27 + 1959353*x^26 - 29225*x^25 - 39423204*x^24 + 40518048*x^23 + 513641811*x^22 - 918583926*x^21 - 4362858468*x^20 + 10893040871*x^19 + 23607826158*x^18 - 79952364670*x^17 - 75116228962*x^16 + 381639679930*x^15 + 98812353215*x^14 - 1205788951741*x^13 + 175885903114*x^12 + 2530475990857*x^11 - 987113141315*x^10 - 3498652499635*x^9 + 1841635007257*x^8 + 3110258033850*x^7 - 1787906886467*x^6 - 1685908305594*x^5 + 902138910730*x^4 + 495599994390*x^3 - 195854103442*x^2 - 57250484608*x + 8460460991)
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)]