# SageMath code for working with number field 42.42.5239206534209069133889646882729090965733830111939046184442828436267483950448112600301.1
# 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^42 - x^41 - 85*x^40 + 85*x^39 + 3355*x^38 - 3355*x^37 - 81613*x^36 + 81613*x^35 + 1369379*x^34 - 1369379*x^33 - 16806205*x^32 + 16806205*x^31 + 156107459*x^30 - 156107459*x^29 - 1120160061*x^28 + 1120160061*x^27 + 6282191555*x^26 - 6282191555*x^25 - 27681539389*x^24 + 27681539389*x^23 + 95822936771*x^22 - 95822936771*x^21 - 259252432189*x^20 + 259252432189*x^19 + 542530659011*x^18 - 542530659011*x^17 - 863673531709*x^16 + 863673531709*x^15 + 1020501541571*x^14 - 1020501541571*x^13 - 863673531709*x^12 + 863673531709*x^11 + 497119576771*x^10 - 497119576771*x^9 - 180198260029*x^8 + 180198260029*x^7 + 36543447747*x^6 - 36543447747*x^5 - 3382656317*x^4 + 3382656317*x^3 + 89178819*x^2 - 89178819*x - 998717)
# 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^42 - x^41 - 85*x^40 + 85*x^39 + 3355*x^38 - 3355*x^37 - 81613*x^36 + 81613*x^35 + 1369379*x^34 - 1369379*x^33 - 16806205*x^32 + 16806205*x^31 + 156107459*x^30 - 156107459*x^29 - 1120160061*x^28 + 1120160061*x^27 + 6282191555*x^26 - 6282191555*x^25 - 27681539389*x^24 + 27681539389*x^23 + 95822936771*x^22 - 95822936771*x^21 - 259252432189*x^20 + 259252432189*x^19 + 542530659011*x^18 - 542530659011*x^17 - 863673531709*x^16 + 863673531709*x^15 + 1020501541571*x^14 - 1020501541571*x^13 - 863673531709*x^12 + 863673531709*x^11 + 497119576771*x^10 - 497119576771*x^9 - 180198260029*x^8 + 180198260029*x^7 + 36543447747*x^6 - 36543447747*x^5 - 3382656317*x^4 + 3382656317*x^3 + 89178819*x^2 - 89178819*x - 998717)
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)]