# SageMath code for working with number field 32.32.8052845212573000012543979797231296934933304854055472857088000000000000000000000000.2 # 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^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250) # 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^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 25046768244100781250) 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)]