# SageMath code for working with number field 27.27.13122249213311579236015222881822854848980675468493365969.5
# 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 - 9*x^26 - 144*x^25 + 1152*x^24 + 8748*x^23 - 53568*x^22 - 299961*x^21 + 1145826*x^20 + 5967756*x^19 - 11122377*x^18 - 64116414*x^17 + 40038543*x^16 + 341380242*x^15 - 21808332*x^14 - 928546470*x^13 - 170651646*x^12 + 1278782856*x^11 + 375962418*x^10 - 839220759*x^9 - 271274076*x^8 + 239523237*x^7 + 68849766*x^6 - 29543454*x^5 - 5065389*x^4 + 1859094*x^3 + 44577*x^2 - 39771*x + 2427)
# 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 - 9*x^26 - 144*x^25 + 1152*x^24 + 8748*x^23 - 53568*x^22 - 299961*x^21 + 1145826*x^20 + 5967756*x^19 - 11122377*x^18 - 64116414*x^17 + 40038543*x^16 + 341380242*x^15 - 21808332*x^14 - 928546470*x^13 - 170651646*x^12 + 1278782856*x^11 + 375962418*x^10 - 839220759*x^9 - 271274076*x^8 + 239523237*x^7 + 68849766*x^6 - 29543454*x^5 - 5065389*x^4 + 1859094*x^3 + 44577*x^2 - 39771*x + 2427)
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)]