# SageMath code for working with number field 27.1.100449472719512733423421001345835601036409912109375.1
# 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^27 - 82*x^24 - 9*x^23 + 195*x^22 + 2387*x^21 + 810*x^20 - 11814*x^19 - 33683*x^18 + 2520*x^17 + 230379*x^16 + 441880*x^15 - 1170738*x^14 - 529926*x^13 - 6335185*x^12 + 26111763*x^11 - 28861455*x^10 + 54201969*x^9 - 172123947*x^8 + 231730104*x^7 - 172770969*x^6 + 271759050*x^5 - 338641350*x^4 + 41402575*x^3 + 177368400*x^2 - 96664125*x + 14269375)
# 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 - 82*x^24 - 9*x^23 + 195*x^22 + 2387*x^21 + 810*x^20 - 11814*x^19 - 33683*x^18 + 2520*x^17 + 230379*x^16 + 441880*x^15 - 1170738*x^14 - 529926*x^13 - 6335185*x^12 + 26111763*x^11 - 28861455*x^10 + 54201969*x^9 - 172123947*x^8 + 231730104*x^7 - 172770969*x^6 + 271759050*x^5 - 338641350*x^4 + 41402575*x^3 + 177368400*x^2 - 96664125*x + 14269375)
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)]