// Magma code for working with number field 38.0.3600319611560698860610362435063272860144872381060102455880973038531255093138465367428016635904.1 // 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^38 + 181*x^36 + 14302*x^34 + 655785*x^32 + 19564842*x^30 + 403491764*x^28 + 5961216274*x^26 + 64460499272*x^24 + 516273473421*x^22 + 3076728313208*x^20 + 13620343868498*x^18 + 44444158242663*x^16 + 105381413708670*x^14 + 177560394917737*x^12 + 205585970435602*x^10 + 155336297697903*x^8 + 70268784621098*x^6 + 16089414412134*x^4 + 1207108975793*x^2 + 13841287201); // 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^38 + 181*x^36 + 14302*x^34 + 655785*x^32 + 19564842*x^30 + 403491764*x^28 + 5961216274*x^26 + 64460499272*x^24 + 516273473421*x^22 + 3076728313208*x^20 + 13620343868498*x^18 + 44444158242663*x^16 + 105381413708670*x^14 + 177560394917737*x^12 + 205585970435602*x^10 + 155336297697903*x^8 + 70268784621098*x^6 + 16089414412134*x^4 + 1207108975793*x^2 + 13841287201); 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))];