// Magma code for working with number field 27.1.86311832016540901180083101912199499488333854262271.1 // Some of these functions may take a long time to execute (this depends on the field). // Define the number field: R := PolynomialRing(Rationals()); K := NumberField(x^27 - 9*x^26 + 51*x^25 - 157*x^24 + 381*x^23 + 94*x^22 - 4772*x^21 + 22385*x^20 - 19633*x^19 - 132957*x^18 + 462384*x^17 + 49228*x^16 - 4729835*x^15 + 16673567*x^14 - 15312382*x^13 - 102305328*x^12 + 361800096*x^11 - 196173035*x^10 - 890969729*x^9 + 1569260108*x^8 - 437613721*x^7 - 1278467664*x^6 + 2134193061*x^5 - 1922367953*x^4 + 1085805378*x^3 - 524572351*x^2 + 152402636*x - 24878621); // Defining polynomial: DefiningPolynomial(K); // Degree over Q: Degree(K); // Signature: Signature(K); // Discriminant: OK := Integers(K); Discriminant(OK); // Ramified primes: PrimeDivisors(Discriminant(OK)); // Autmorphisms: Automorphisms(K); // Integral basis: IntegralBasis(K); // Class group: ClassGroup(K); // Unit group: UK, fUK := UnitGroup(K); // Unit rank: UnitRank(K); // Generator for roots of unity: K!f(TU.1) where TU,f is TorsionUnitGroup(K); // Fundamental units: [K|fUK(g): g in Generators(UK)]; // Regulator: Regulator(K); // Analytic class number formula: /* self-contained Magma code snippet to compute the analytic class number formula */ Qx := PolynomialRing(QQ); K := NumberField(x^27 - 9*x^26 + 51*x^25 - 157*x^24 + 381*x^23 + 94*x^22 - 4772*x^21 + 22385*x^20 - 19633*x^19 - 132957*x^18 + 462384*x^17 + 49228*x^16 - 4729835*x^15 + 16673567*x^14 - 15312382*x^13 - 102305328*x^12 + 361800096*x^11 - 196173035*x^10 - 890969729*x^9 + 1569260108*x^8 - 437613721*x^7 - 1278467664*x^6 + 2134193061*x^5 - 1922367953*x^4 + 1085805378*x^3 - 524572351*x^2 + 152402636*x - 24878621); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK))); // Intermediate fields: L := Subfields(K); L[2..#L]; // Galois group: G = GaloisGroup(K); // 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 Magma: p := 7; [ : pr in Factorization(p*Integers(K))];