# SageMath code for working with number field 27.27.33967771186402966366427549984621879030979398934601324527479889.6
# 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^27 - 198*x^25 - 240*x^24 + 14211*x^23 + 27828*x^22 - 470235*x^21 - 1095210*x^20 + 7805736*x^19 + 17721963*x^18 - 75980394*x^17 - 145128888*x^16 + 462567705*x^15 + 653137839*x^14 - 1750939695*x^13 - 1685642442*x^12 + 4004759304*x^11 + 2575237032*x^10 - 5330234694*x^9 - 2406041811*x^8 + 3884585238*x^7 + 1396307361*x^6 - 1366755093*x^5 - 467018919*x^4 + 163985460*x^3 + 65743461*x^2 + 1605852*x - 707723)
# 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^27 - 198*x^25 - 240*x^24 + 14211*x^23 + 27828*x^22 - 470235*x^21 - 1095210*x^20 + 7805736*x^19 + 17721963*x^18 - 75980394*x^17 - 145128888*x^16 + 462567705*x^15 + 653137839*x^14 - 1750939695*x^13 - 1685642442*x^12 + 4004759304*x^11 + 2575237032*x^10 - 5330234694*x^9 - 2406041811*x^8 + 3884585238*x^7 + 1396307361*x^6 - 1366755093*x^5 - 467018919*x^4 + 163985460*x^3 + 65743461*x^2 + 1605852*x - 707723)
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)]