# SageMath code for working with number field 27.1.19977770457280786686246139734781554667936251692291.1
# 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 - 2*x^26 + 16*x^25 + 62*x^24 + 342*x^23 + 1648*x^22 + 6094*x^21 + 15834*x^20 + 34184*x^19 + 61822*x^18 + 112334*x^17 + 237408*x^16 + 637440*x^15 + 1828206*x^14 + 5099608*x^13 + 12463122*x^12 + 26430338*x^11 + 48978608*x^10 + 80603954*x^9 + 116977534*x^8 + 149538616*x^7 + 167922122*x^6 + 164525402*x^5 + 137815936*x^4 + 94764543*x^3 + 50090044*x^2 + 17954520*x + 3418472)
# 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 - 2*x^26 + 16*x^25 + 62*x^24 + 342*x^23 + 1648*x^22 + 6094*x^21 + 15834*x^20 + 34184*x^19 + 61822*x^18 + 112334*x^17 + 237408*x^16 + 637440*x^15 + 1828206*x^14 + 5099608*x^13 + 12463122*x^12 + 26430338*x^11 + 48978608*x^10 + 80603954*x^9 + 116977534*x^8 + 149538616*x^7 + 167922122*x^6 + 164525402*x^5 + 137815936*x^4 + 94764543*x^3 + 50090044*x^2 + 17954520*x + 3418472)
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)]