# SageMath code for working with number field 42.0.959396304051793463814262846982490027578741814649477038563926538598268329263104.1 # 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^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1) # 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^42 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 1) 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)]