# SageMath code for working with number field 33.11.1635170022196481349560959748587682926364327.1
# Some of these functions may take a long time to execute (this depends on the field).
# Define the number field:
x = polygen(QQ); K. = NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*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. = NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*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)]