# SageMath code for working with number field 27.27.138518475817966243726955928755744937608694859226091439662129969.7
# 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 - 378*x^25 - 180*x^24 + 55404*x^23 + 24111*x^22 - 4270554*x^21 - 1205037*x^20 + 193978638*x^19 + 34664778*x^18 - 5500402929*x^17 - 826631784*x^16 + 100060233273*x^15 + 17341462773*x^14 - 1173646133964*x^13 - 242321726520*x^12 + 8757504551097*x^11 + 1781259610428*x^10 - 40408055580087*x^9 - 5139431874873*x^8 + 110255393136807*x^7 - 2797660463697*x^6 - 161409309518763*x^5 + 35414591654925*x^4 + 96140161943631*x^3 - 36022701760533*x^2 - 4256570175237*x - 76637740149)
# 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 - 378*x^25 - 180*x^24 + 55404*x^23 + 24111*x^22 - 4270554*x^21 - 1205037*x^20 + 193978638*x^19 + 34664778*x^18 - 5500402929*x^17 - 826631784*x^16 + 100060233273*x^15 + 17341462773*x^14 - 1173646133964*x^13 - 242321726520*x^12 + 8757504551097*x^11 + 1781259610428*x^10 - 40408055580087*x^9 - 5139431874873*x^8 + 110255393136807*x^7 - 2797660463697*x^6 - 161409309518763*x^5 + 35414591654925*x^4 + 96140161943631*x^3 - 36022701760533*x^2 - 4256570175237*x - 76637740149)
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)]