// Magma code for working with number field 31.31.303180754947920226773910663600827932461415650100367091342054488471568688197420529491484801.1 // 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^31 - 465*x^29 - 341*x^28 + 91078*x^27 + 119908*x^26 - 9914451*x^25 - 18008892*x^24 + 666449563*x^23 + 1538774528*x^22 - 28934386579*x^21 - 83032233092*x^20 + 817351924162*x^19 + 2943382779538*x^18 - 14471455469735*x^17 - 68751097280499*x^16 + 137645143880654*x^15 + 1028543867187214*x^14 - 144266178422457*x^13 - 9110738641752719*x^12 - 11199776062831713*x^11 + 37846129267859005*x^10 + 105950708850677777*x^9 + 9682358886535270*x^8 - 290542929220278846*x^7 - 437492517497239584*x^6 - 201727405103545942*x^5 + 73256081844015542*x^4 + 87563225965427030*x^3 + 7669633084320922*x^2 - 7522164033404415*x - 310695313260929); // 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^31 - 465*x^29 - 341*x^28 + 91078*x^27 + 119908*x^26 - 9914451*x^25 - 18008892*x^24 + 666449563*x^23 + 1538774528*x^22 - 28934386579*x^21 - 83032233092*x^20 + 817351924162*x^19 + 2943382779538*x^18 - 14471455469735*x^17 - 68751097280499*x^16 + 137645143880654*x^15 + 1028543867187214*x^14 - 144266178422457*x^13 - 9110738641752719*x^12 - 11199776062831713*x^11 + 37846129267859005*x^10 + 105950708850677777*x^9 + 9682358886535270*x^8 - 290542929220278846*x^7 - 437492517497239584*x^6 - 201727405103545942*x^5 + 73256081844015542*x^4 + 87563225965427030*x^3 + 7669633084320922*x^2 - 7522164033404415*x - 310695313260929); 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))];