# Oscar code for working with number field 28.28.18917407352603402612306290142504402709970002327473311711232.1 # 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 = PolynomialRing(QQ); K, a = NumberField(x^28 - 4*x^27 - 70*x^26 + 272*x^25 + 2007*x^24 - 7620*x^23 - 31106*x^22 + 115948*x^21 + 290042*x^20 - 1063528*x^19 - 1718136*x^18 + 6176908*x^17 + 6666823*x^16 - 23204920*x^15 - 17269984*x^14 + 56540840*x^13 + 30251654*x^12 - 88343908*x^11 - 35706314*x^10 + 86510324*x^9 + 26775430*x^8 - 51247828*x^7 - 10970104*x^6 + 17598492*x^5 + 1821397*x^4 - 3152012*x^3 + 46562*x^2 + 211268*x - 23953) # 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^28 - 4*x^27 - 70*x^26 + 272*x^25 + 2007*x^24 - 7620*x^23 - 31106*x^22 + 115948*x^21 + 290042*x^20 - 1063528*x^19 - 1718136*x^18 + 6176908*x^17 + 6666823*x^16 - 23204920*x^15 - 17269984*x^14 + 56540840*x^13 + 30251654*x^12 - 88343908*x^11 - 35706314*x^10 + 86510324*x^9 + 26775430*x^8 - 51247828*x^7 - 10970104*x^6 + 17598492*x^5 + 1821397*x^4 - 3152012*x^3 + 46562*x^2 + 211268*x - 23953); 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]