# SageMath code for working with number field 23.23.542693874230042671882983450092579839839306394717839529.1
# 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^23 - x^22 - 132*x^21 + 269*x^20 + 6722*x^19 - 19118*x^18 - 167488*x^17 + 601226*x^16 + 2127300*x^15 - 9664345*x^14 - 12698450*x^13 + 83446410*x^12 + 19888027*x^11 - 390244377*x^10 + 110929757*x^9 + 994369765*x^8 - 511651057*x^7 - 1370318108*x^6 + 775297206*x^5 + 972267257*x^4 - 466392148*x^3 - 312747839*x^2 + 85444683*x + 38737459)
# 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^23 - x^22 - 132*x^21 + 269*x^20 + 6722*x^19 - 19118*x^18 - 167488*x^17 + 601226*x^16 + 2127300*x^15 - 9664345*x^14 - 12698450*x^13 + 83446410*x^12 + 19888027*x^11 - 390244377*x^10 + 110929757*x^9 + 994369765*x^8 - 511651057*x^7 - 1370318108*x^6 + 775297206*x^5 + 972267257*x^4 - 466392148*x^3 - 312747839*x^2 + 85444683*x + 38737459)
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)]