# SageMath code for working with number field 29.1.57471868152223727924656865491755923007187161136129.1
# 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^29 - x^28 + 18*x^27 + 12*x^26 + 179*x^25 + 123*x^24 + 1662*x^23 - 196*x^22 + 11119*x^21 - 9104*x^20 + 47466*x^19 - 44547*x^18 + 83114*x^17 - 23886*x^16 - 121329*x^15 + 248495*x^14 - 275546*x^13 - 188738*x^12 + 573921*x^11 - 476142*x^10 + 379865*x^9 - 32019*x^8 + 20658*x^7 - 6163*x^6 - 27930*x^5 + 2012*x^4 - 1404*x^3 + 2898*x^2 - 594*x + 81)
# 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^29 - x^28 + 18*x^27 + 12*x^26 + 179*x^25 + 123*x^24 + 1662*x^23 - 196*x^22 + 11119*x^21 - 9104*x^20 + 47466*x^19 - 44547*x^18 + 83114*x^17 - 23886*x^16 - 121329*x^15 + 248495*x^14 - 275546*x^13 - 188738*x^12 + 573921*x^11 - 476142*x^10 + 379865*x^9 - 32019*x^8 + 20658*x^7 - 6163*x^6 - 27930*x^5 + 2012*x^4 - 1404*x^3 + 2898*x^2 - 594*x + 81)
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)]