# Oscar code for working with number field 27.27.4737131966005480104201495460338050600482023844126105114809.4 # 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 - 198*x^25 - 234*x^24 + 15957*x^23 + 34857*x^22 - 664605*x^21 - 2060424*x^20 + 14909967*x^19 + 61408524*x^18 - 164914623*x^17 - 969682860*x^16 + 509316831*x^15 + 7881823827*x^14 + 4987250568*x^13 - 30694701609*x^12 - 45067675860*x^11 + 41145793425*x^10 + 121189746357*x^9 + 34092403551*x^8 - 95018229873*x^7 - 74437098228*x^6 + 10022089581*x^5 + 25924612491*x^4 + 5241208824*x^3 - 2001751137*x^2 - 781144074*x - 68912373) # 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 - 198*x^25 - 234*x^24 + 15957*x^23 + 34857*x^22 - 664605*x^21 - 2060424*x^20 + 14909967*x^19 + 61408524*x^18 - 164914623*x^17 - 969682860*x^16 + 509316831*x^15 + 7881823827*x^14 + 4987250568*x^13 - 30694701609*x^12 - 45067675860*x^11 + 41145793425*x^10 + 121189746357*x^9 + 34092403551*x^8 - 95018229873*x^7 - 74437098228*x^6 + 10022089581*x^5 + 25924612491*x^4 + 5241208824*x^3 - 2001751137*x^2 - 781144074*x - 68912373); 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]