# SageMath code for working with number field 27.27.13122249213311579236015222881822854848980675468493365969.5 # 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 - 9*x^26 - 144*x^25 + 1152*x^24 + 8748*x^23 - 53568*x^22 - 299961*x^21 + 1145826*x^20 + 5967756*x^19 - 11122377*x^18 - 64116414*x^17 + 40038543*x^16 + 341380242*x^15 - 21808332*x^14 - 928546470*x^13 - 170651646*x^12 + 1278782856*x^11 + 375962418*x^10 - 839220759*x^9 - 271274076*x^8 + 239523237*x^7 + 68849766*x^6 - 29543454*x^5 - 5065389*x^4 + 1859094*x^3 + 44577*x^2 - 39771*x + 2427) # 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 - 9*x^26 - 144*x^25 + 1152*x^24 + 8748*x^23 - 53568*x^22 - 299961*x^21 + 1145826*x^20 + 5967756*x^19 - 11122377*x^18 - 64116414*x^17 + 40038543*x^16 + 341380242*x^15 - 21808332*x^14 - 928546470*x^13 - 170651646*x^12 + 1278782856*x^11 + 375962418*x^10 - 839220759*x^9 - 271274076*x^8 + 239523237*x^7 + 68849766*x^6 - 29543454*x^5 - 5065389*x^4 + 1859094*x^3 + 44577*x^2 - 39771*x + 2427) 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)]