// Magma code for working with number field 30.2.51586587308205615747892932945519983768463134765625.1 // 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^30 - 10*x^29 + 36*x^28 - 34*x^27 - 129*x^26 + 673*x^25 - 2345*x^24 + 3400*x^23 + 8195*x^22 - 23985*x^21 - 13829*x^20 + 36240*x^19 + 94321*x^18 + 102411*x^17 + 62946*x^16 + 325659*x^15 + 1299805*x^14 + 3826146*x^13 + 7335516*x^12 + 9580506*x^11 + 8499204*x^10 + 5247290*x^9 + 1937111*x^8 + 388946*x^7 - 47819*x^6 - 74039*x^5 - 43685*x^4 - 5057*x^3 + 3493*x^2 + 123*x + 169); // 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^30 - 10*x^29 + 36*x^28 - 34*x^27 - 129*x^26 + 673*x^25 - 2345*x^24 + 3400*x^23 + 8195*x^22 - 23985*x^21 - 13829*x^20 + 36240*x^19 + 94321*x^18 + 102411*x^17 + 62946*x^16 + 325659*x^15 + 1299805*x^14 + 3826146*x^13 + 7335516*x^12 + 9580506*x^11 + 8499204*x^10 + 5247290*x^9 + 1937111*x^8 + 388946*x^7 - 47819*x^6 - 74039*x^5 - 43685*x^4 - 5057*x^3 + 3493*x^2 + 123*x + 169); 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))];