// Magma code for working with number field 30.30.5950661074415937716058277355262049126611998411687341.1 // (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, -15, 120, 560, -2380, -6188, 18564, 31824, -75582, -92378, 184756, 167960, -293930, -203490, 319770, 170544, -245157, -100947, 134596, 42504, -53130, -12650, 14950, 2600, -2925, -351, 378, 28, -29, -1, 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];