# SageMath code for working with number field 32.0.1156745857256138299057661301456078800598048178176.1
# 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 + 10*x^30 - 28*x^29 + 56*x^28 - 208*x^27 + 604*x^26 - 876*x^25 + 2547*x^24 - 5760*x^23 + 7684*x^22 - 11384*x^21 + 37676*x^20 - 21776*x^19 + 12262*x^18 - 135456*x^17 + 96553*x^16 + 443172*x^15 + 1107228*x^14 + 758160*x^13 - 662400*x^12 - 2309040*x^11 - 1681344*x^10 + 1407456*x^9 + 4596912*x^8 + 3872448*x^7 - 62208*x^6 - 4385664*x^5 - 3545856*x^4 + 3919104*x^2 + 3359232*x + 1679616)
# 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 + 10*x^30 - 28*x^29 + 56*x^28 - 208*x^27 + 604*x^26 - 876*x^25 + 2547*x^24 - 5760*x^23 + 7684*x^22 - 11384*x^21 + 37676*x^20 - 21776*x^19 + 12262*x^18 - 135456*x^17 + 96553*x^16 + 443172*x^15 + 1107228*x^14 + 758160*x^13 - 662400*x^12 - 2309040*x^11 - 1681344*x^10 + 1407456*x^9 + 4596912*x^8 + 3872448*x^7 - 62208*x^6 - 4385664*x^5 - 3545856*x^4 + 3919104*x^2 + 3359232*x + 1679616)
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)]