# SageMath code for working with number field 27.27.138518475817966243726955928755744937608694859226091439662129969.7 # 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^27 - 378*x^25 - 180*x^24 + 55404*x^23 + 24111*x^22 - 4270554*x^21 - 1205037*x^20 + 193978638*x^19 + 34664778*x^18 - 5500402929*x^17 - 826631784*x^16 + 100060233273*x^15 + 17341462773*x^14 - 1173646133964*x^13 - 242321726520*x^12 + 8757504551097*x^11 + 1781259610428*x^10 - 40408055580087*x^9 - 5139431874873*x^8 + 110255393136807*x^7 - 2797660463697*x^6 - 161409309518763*x^5 + 35414591654925*x^4 + 96140161943631*x^3 - 36022701760533*x^2 - 4256570175237*x - 76637740149) # 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^27 - 378*x^25 - 180*x^24 + 55404*x^23 + 24111*x^22 - 4270554*x^21 - 1205037*x^20 + 193978638*x^19 + 34664778*x^18 - 5500402929*x^17 - 826631784*x^16 + 100060233273*x^15 + 17341462773*x^14 - 1173646133964*x^13 - 242321726520*x^12 + 8757504551097*x^11 + 1781259610428*x^10 - 40408055580087*x^9 - 5139431874873*x^8 + 110255393136807*x^7 - 2797660463697*x^6 - 161409309518763*x^5 + 35414591654925*x^4 + 96140161943631*x^3 - 36022701760533*x^2 - 4256570175237*x - 76637740149) 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)]