# SageMath code for working with number field 42.0.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1
# 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 + 84*x^40 + 3276*x^38 + 78736*x^36 + 1305360*x^34 + 15833664*x^32 + 145435136*x^30 + 1032886400*x^28 + 5741639680*x^26 + 25131904000*x^24 + 86707088384*x^22 + 234966480896*x^20 + 496154537984*x^18 + 805934137344*x^16 + 988445753344*x^14 + 891877195776*x^12 + 571258175488*x^10 + 247113187328*x^8 + 67166797824*x^6 + 10250354688*x^4 + 719323136*x^2 + 14680064)
# 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 + 84*x^40 + 3276*x^38 + 78736*x^36 + 1305360*x^34 + 15833664*x^32 + 145435136*x^30 + 1032886400*x^28 + 5741639680*x^26 + 25131904000*x^24 + 86707088384*x^22 + 234966480896*x^20 + 496154537984*x^18 + 805934137344*x^16 + 988445753344*x^14 + 891877195776*x^12 + 571258175488*x^10 + 247113187328*x^8 + 67166797824*x^6 + 10250354688*x^4 + 719323136*x^2 + 14680064)
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)]