# SageMath code for working with number field 38.0.3600319611560698860610362435063272860144872381060102455880973038531255093138465367428016635904.1
# 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^38 + 181*x^36 + 14302*x^34 + 655785*x^32 + 19564842*x^30 + 403491764*x^28 + 5961216274*x^26 + 64460499272*x^24 + 516273473421*x^22 + 3076728313208*x^20 + 13620343868498*x^18 + 44444158242663*x^16 + 105381413708670*x^14 + 177560394917737*x^12 + 205585970435602*x^10 + 155336297697903*x^8 + 70268784621098*x^6 + 16089414412134*x^4 + 1207108975793*x^2 + 13841287201)
# 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^38 + 181*x^36 + 14302*x^34 + 655785*x^32 + 19564842*x^30 + 403491764*x^28 + 5961216274*x^26 + 64460499272*x^24 + 516273473421*x^22 + 3076728313208*x^20 + 13620343868498*x^18 + 44444158242663*x^16 + 105381413708670*x^14 + 177560394917737*x^12 + 205585970435602*x^10 + 155336297697903*x^8 + 70268784621098*x^6 + 16089414412134*x^4 + 1207108975793*x^2 + 13841287201)
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)]