# SageMath code for working with number field 33.33.70011645999218458416472683122408534303895571350166174758601569.1

# 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 - 63*x^31 - 4*x^30 + 1710*x^29 + 204*x^28 - 26254*x^27 - 4392*x^26 + 251754*x^25 + 52248*x^24 - 1572192*x^23 - 377544*x^22 + 6479940*x^21 + 1717920*x^20 - 17548167*x^19 - 4953616*x^18 + 30725595*x^17 + 8949468*x^16 - 34072421*x^15 - 9933084*x^14 + 23552382*x^13 + 6584132*x^12 - 10086444*x^11 - 2580438*x^10 + 2636960*x^9 + 587106*x^8 - 404862*x^7 - 74061*x^6 + 33768*x^5 + 4755*x^4 - 1326*x^3 - 135*x^2 + 18*x + 1)

# 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 - 63*x^31 - 4*x^30 + 1710*x^29 + 204*x^28 - 26254*x^27 - 4392*x^26 + 251754*x^25 + 52248*x^24 - 1572192*x^23 - 377544*x^22 + 6479940*x^21 + 1717920*x^20 - 17548167*x^19 - 4953616*x^18 + 30725595*x^17 + 8949468*x^16 - 34072421*x^15 - 9933084*x^14 + 23552382*x^13 + 6584132*x^12 - 10086444*x^11 - 2580438*x^10 + 2636960*x^9 + 587106*x^8 - 404862*x^7 - 74061*x^6 + 33768*x^5 + 4755*x^4 - 1326*x^3 - 135*x^2 + 18*x + 1)
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)]