# Oscar code for working with number field 27.1.1543319746516623033280478216838436483146079.1 # If you have not already loaded the Oscar package, you should type "using Oscar;" before running the code below. # Some of these functions may take a long time to compile (this depends on the state of your Julia REPL), and/or to execute (this depends on the field). # Define the number field: Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1) # Defining polynomial: defining_polynomial(K) # Degree over Q: degree(K) # Signature: signature(K) # Discriminant: OK = ring_of_integers(K); discriminant(OK) # Ramified primes: prime_divisors(discriminant((OK))) # Autmorphisms: automorphisms(K) # Integral basis: basis(OK) # Class group: class_group(K) # Unit group: UK, fUK = unit_group(OK) # Unit rank: rank(UK) # Generator for roots of unity: torsion_units_generator(OK) # Fundamental units: [K(fUK(a)) for a in gens(UK)] # Regulator: regulator(K) # Analytic class number formula: # self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK)))) # Intermediate fields: subfields(K)[2:end-1] # Galois group: G, Gtx = galois_group(K); G, transitive_group_identification(G) # 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 Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]