# 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)]