# SageMath code for working with number field 27.1.149674927005884133619412112407487159468032.1 # 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^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80) # 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 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80) 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)]