# SageMath code for working with number field 33.33.277966181338944111003326058293667039541136678070715028736001.1
# 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^33 - x^32 - 52*x^31 + 47*x^30 + 1159*x^29 - 945*x^28 - 14589*x^27 + 10741*x^26 + 115099*x^25 - 76770*x^24 - 597580*x^23 + 362822*x^22 + 2088771*x^21 - 1160546*x^20 - 4956062*x^19 + 2531215*x^18 + 7980244*x^17 - 3753538*x^16 - 8674935*x^15 + 3750773*x^14 + 6304834*x^13 - 2497276*x^12 - 3006793*x^11 + 1087323*x^10 + 908614*x^9 - 298518*x^8 - 163557*x^7 + 48112*x^6 + 15777*x^5 - 3955*x^4 - 691*x^3 + 126*x^2 + 12*x - 1)
# 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^33 - x^32 - 52*x^31 + 47*x^30 + 1159*x^29 - 945*x^28 - 14589*x^27 + 10741*x^26 + 115099*x^25 - 76770*x^24 - 597580*x^23 + 362822*x^22 + 2088771*x^21 - 1160546*x^20 - 4956062*x^19 + 2531215*x^18 + 7980244*x^17 - 3753538*x^16 - 8674935*x^15 + 3750773*x^14 + 6304834*x^13 - 2497276*x^12 - 3006793*x^11 + 1087323*x^10 + 908614*x^9 - 298518*x^8 - 163557*x^7 + 48112*x^6 + 15777*x^5 - 3955*x^4 - 691*x^3 + 126*x^2 + 12*x - 1)
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)]