// Magma code for working with number field 27.27.13122249213311579236015222881822854848980675468493365969.2 // 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 - 144*x^25 + 1314*x^24 + 7740*x^23 - 68076*x^22 - 239166*x^21 + 1714032*x^20 + 4990779*x^19 - 22424565*x^18 - 67949397*x^17 + 139167009*x^16 + 521720982*x^15 - 259125408*x^14 - 1876452975*x^13 - 322111764*x^12 + 3304120158*x^11 + 1504663875*x^10 - 3180640551*x^9 - 1866753171*x^8 + 1770577101*x^7 + 1116671832*x^6 - 573082407*x^5 - 338304879*x^4 + 102031929*x^3 + 44766450*x^2 - 8037450*x - 1183123); // 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 - 144*x^25 + 1314*x^24 + 7740*x^23 - 68076*x^22 - 239166*x^21 + 1714032*x^20 + 4990779*x^19 - 22424565*x^18 - 67949397*x^17 + 139167009*x^16 + 521720982*x^15 - 259125408*x^14 - 1876452975*x^13 - 322111764*x^12 + 3304120158*x^11 + 1504663875*x^10 - 3180640551*x^9 - 1866753171*x^8 + 1770577101*x^7 + 1116671832*x^6 - 573082407*x^5 - 338304879*x^4 + 102031929*x^3 + 44766450*x^2 - 8037450*x - 1183123); 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))];