// Magma code for working with number field 30.0.159386923550435671074967363509984324121230045171.1 // (Note that not all these functions may be available, and some may take a long time to execute.) // Define the number field: R := PolynomialRing(Rationals()); K := NumberField(R![1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 10314, 16093, 5523, 6255, 2672, 2493, 1224, 1000, 507, 297, -109, 90, -27, 28, -6, 9, -1, 3, 0, 1]); // Defining polynomial: DefiningPolynomial(K); // Degree over Q: Degree(K); // Signature: Signature(K); // Discriminant: Discriminant(Integers(K)); // Ramified primes: PrimeDivisors(Discriminant(Integers(K))); // Integral basis: IntegralBasis(K); // Class group: ClassGroup(K); // Unit group: UK, f := UnitGroup(K); // Unit rank: UnitRank(K); // Generator for roots of unity: K!f(TU.1) where TU,f is TorsionUnitGroup(K); // Fundamental units: [K!f(g): g in Generators(UK)]; // Regulator: Regulator(K); // Galois group: GaloisGroup(K); // Frobenius cycle types: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$: idealfactors := Factorization(p*Integers(K)); // get the data [ : primefactor in idealfactors];