// Magma code for working with number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.1 // 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^44 - x^43 + 43*x^42 - 39*x^41 + 856*x^40 - 700*x^39 + 10478*x^38 - 7678*x^37 + 88356*x^36 - 57644*x^35 + 545008*x^34 - 314432*x^33 + 2548537*x^32 - 1290809*x^31 + 9238279*x^30 - 4075043*x^29 + 26320891*x^28 - 10020719*x^27 + 59401115*x^26 - 19318239*x^25 + 106508095*x^24 - 29235139*x^23 + 151556799*x^22 - 34552164*x^21 + 170215728*x^20 - 30533255*x^19 + 148629623*x^18 - 11758433*x^17 + 94928616*x^16 + 36112685*x^15 + 30135721*x^14 + 126072207*x^13 - 23633656*x^12 + 220881720*x^11 - 44029406*x^10 + 233441302*x^9 - 35764283*x^8 + 155778230*x^7 - 8938748*x^6 + 18093321*x^5 + 18069001*x^4 + 120313546*x^3 - 192566274*x^2 - 256263936*x + 1026529561); // 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^44 - x^43 + 43*x^42 - 39*x^41 + 856*x^40 - 700*x^39 + 10478*x^38 - 7678*x^37 + 88356*x^36 - 57644*x^35 + 545008*x^34 - 314432*x^33 + 2548537*x^32 - 1290809*x^31 + 9238279*x^30 - 4075043*x^29 + 26320891*x^28 - 10020719*x^27 + 59401115*x^26 - 19318239*x^25 + 106508095*x^24 - 29235139*x^23 + 151556799*x^22 - 34552164*x^21 + 170215728*x^20 - 30533255*x^19 + 148629623*x^18 - 11758433*x^17 + 94928616*x^16 + 36112685*x^15 + 30135721*x^14 + 126072207*x^13 - 23633656*x^12 + 220881720*x^11 - 44029406*x^10 + 233441302*x^9 - 35764283*x^8 + 155778230*x^7 - 8938748*x^6 + 18093321*x^5 + 18069001*x^4 + 120313546*x^3 - 192566274*x^2 - 256263936*x + 1026529561); 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))];