# SageMath code for working with number field 42.0.1236405605609949863710440275882328463150065157474288679507165498832965910232619214817863.1
# 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 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417)
# 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 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417)
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)]