# SageMath code for working with number field 33.33.19453503693937104587930129781385805482525288081047228454446873116938739522375281.1.
# 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 - 108*x^31 + 826*x^30 + 5517*x^29 - 37410*x^28 - 174850*x^27 + 974941*x^26 + 3753002*x^25 - 16170687*x^24 - 56209081*x^23 + 178946863*x^22 + 592568182*x^21 - 1351731560*x^20 - 4397140947*x^19 + 7038388451*x^18 + 22835380012*x^17 - 25350973354*x^16 - 82027847149*x^15 + 63440275941*x^14 + 199922687695*x^13 - 111827183164*x^12 - 320812557227*x^11 + 142179324189*x^10 + 323210228406*x^9 - 130110920902*x^8 - 188148415319*x^7 + 77322457297*x^6 + 52579210647*x^5 - 22741173483*x^4 - 3948105037*x^3 + 1212304894*x^2 + 227068790*x + 8672509)

# 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 - 108*x^31 + 826*x^30 + 5517*x^29 - 37410*x^28 - 174850*x^27 + 974941*x^26 + 3753002*x^25 - 16170687*x^24 - 56209081*x^23 + 178946863*x^22 + 592568182*x^21 - 1351731560*x^20 - 4397140947*x^19 + 7038388451*x^18 + 22835380012*x^17 - 25350973354*x^16 - 82027847149*x^15 + 63440275941*x^14 + 199922687695*x^13 - 111827183164*x^12 - 320812557227*x^11 + 142179324189*x^10 + 323210228406*x^9 - 130110920902*x^8 - 188148415319*x^7 + 77322457297*x^6 + 52579210647*x^5 - 22741173483*x^4 - 3948105037*x^3 + 1212304894*x^2 + 227068790*x + 8672509)
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 pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]