// Magma code for working with number field 32.32.1674417821664703472295706925363164706188610888296487588786531799142113.1 // 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^32 - x^31 - 95*x^30 - 37*x^29 + 3924*x^28 + 6404*x^27 - 86885*x^26 - 247839*x^25 + 1027629*x^24 + 4686255*x^23 - 4671359*x^22 - 48952218*x^21 - 29135620*x^20 + 277610329*x^19 + 503619218*x^18 - 666000824*x^17 - 2706325012*x^16 - 835566428*x^15 + 6403713290*x^14 + 8233352408*x^13 - 3964294981*x^12 - 15570255798*x^11 - 7847282396*x^10 + 8342709494*x^9 + 10369480703*x^8 + 994686632*x^7 - 3490824231*x^6 - 1377789078*x^5 + 354398962*x^4 + 243218618*x^3 - 2768445*x^2 - 9898412*x + 705613); // 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^32 - x^31 - 95*x^30 - 37*x^29 + 3924*x^28 + 6404*x^27 - 86885*x^26 - 247839*x^25 + 1027629*x^24 + 4686255*x^23 - 4671359*x^22 - 48952218*x^21 - 29135620*x^20 + 277610329*x^19 + 503619218*x^18 - 666000824*x^17 - 2706325012*x^16 - 835566428*x^15 + 6403713290*x^14 + 8233352408*x^13 - 3964294981*x^12 - 15570255798*x^11 - 7847282396*x^10 + 8342709494*x^9 + 10369480703*x^8 + 994686632*x^7 - 3490824231*x^6 - 1377789078*x^5 + 354398962*x^4 + 243218618*x^3 - 2768445*x^2 - 9898412*x + 705613); 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))];