# SageMath code for working with number field 32.0.38187227411894497653750080270198291885967163238525390625.1
# 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^32 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*x + 256)
# 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^32 - 2*x^31 - 20*x^30 + 24*x^29 + 286*x^28 - 250*x^27 - 2682*x^26 + 1817*x^25 + 19614*x^24 - 14252*x^23 - 106581*x^22 + 99261*x^21 + 427695*x^20 - 593179*x^19 - 962589*x^18 + 2289866*x^17 - 85839*x^16 - 3519833*x^15 + 7206071*x^14 - 1530693*x^13 + 140965*x^12 + 5408365*x^11 + 271976*x^10 + 335065*x^9 + 2939829*x^8 - 637412*x^7 + 550944*x^6 + 18112*x^5 + 35760*x^4 + 7872*x^3 + 4288*x^2 + 384*x + 256)
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)]