# SageMath code for working with number field 27.27.33967771186402966366427549984621879030979398934601324527479889.6 # 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 - 198*x^25 - 240*x^24 + 14211*x^23 + 27828*x^22 - 470235*x^21 - 1095210*x^20 + 7805736*x^19 + 17721963*x^18 - 75980394*x^17 - 145128888*x^16 + 462567705*x^15 + 653137839*x^14 - 1750939695*x^13 - 1685642442*x^12 + 4004759304*x^11 + 2575237032*x^10 - 5330234694*x^9 - 2406041811*x^8 + 3884585238*x^7 + 1396307361*x^6 - 1366755093*x^5 - 467018919*x^4 + 163985460*x^3 + 65743461*x^2 + 1605852*x - 707723) # 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 - 198*x^25 - 240*x^24 + 14211*x^23 + 27828*x^22 - 470235*x^21 - 1095210*x^20 + 7805736*x^19 + 17721963*x^18 - 75980394*x^17 - 145128888*x^16 + 462567705*x^15 + 653137839*x^14 - 1750939695*x^13 - 1685642442*x^12 + 4004759304*x^11 + 2575237032*x^10 - 5330234694*x^9 - 2406041811*x^8 + 3884585238*x^7 + 1396307361*x^6 - 1366755093*x^5 - 467018919*x^4 + 163985460*x^3 + 65743461*x^2 + 1605852*x - 707723) 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)]