# Oscar code for working with number field 27.27.4038857868055272156360487190465637072631786847797281.1 # 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 - 117*x^25 - 144*x^24 + 5454*x^23 + 12321*x^22 - 127080*x^21 - 415908*x^20 + 1499193*x^19 + 7118643*x^18 - 6881679*x^17 - 66517326*x^16 - 25844319*x^15 + 338054688*x^14 + 452971134*x^13 - 806211819*x^12 - 2101675842*x^11 + 72795897*x^10 + 4152980271*x^9 + 3389225814*x^8 - 2166315039*x^7 - 4727639052*x^6 - 2245523607*x^5 + 227410506*x^4 + 469499994*x^3 + 82966464*x^2 - 18766224*x - 4931621) # 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 - 117*x^25 - 144*x^24 + 5454*x^23 + 12321*x^22 - 127080*x^21 - 415908*x^20 + 1499193*x^19 + 7118643*x^18 - 6881679*x^17 - 66517326*x^16 - 25844319*x^15 + 338054688*x^14 + 452971134*x^13 - 806211819*x^12 - 2101675842*x^11 + 72795897*x^10 + 4152980271*x^9 + 3389225814*x^8 - 2166315039*x^7 - 4727639052*x^6 - 2245523607*x^5 + 227410506*x^4 + 469499994*x^3 + 82966464*x^2 - 18766224*x - 4931621); 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]