# SageMath code for working with number field 45.45.202328376069000568231882298692586415768387771371083469740034658547467305567124640219844877719879150390625.1
# 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^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757)
# 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^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757)
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)]