// Magma code for working with number field 35.35.103338030412840513192336580932106187652481378569380154885948286391794681549072265625.1 // 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^35 - 5*x^34 - 120*x^33 + 560*x^32 + 6265*x^31 - 27063*x^30 - 188895*x^29 + 746800*x^28 + 3677845*x^27 - 13130310*x^26 - 48923530*x^25 + 155517810*x^24 + 458129250*x^23 - 1279983515*x^22 - 3063191465*x^21 + 7446901792*x^20 + 14674072475*x^19 - 30859715040*x^18 - 50096318770*x^17 + 91037787910*x^16 + 120155286235*x^15 - 189447334860*x^14 - 197298653000*x^13 + 272612081065*x^12 + 212356717105*x^11 - 261877354307*x^10 - 139209319745*x^9 + 158110983885*x^8 + 48766657725*x^7 - 53789658890*x^6 - 7150843552*x^5 + 8233276160*x^4 + 456884545*x^3 - 363355535*x^2 - 50594420*x - 1782107); // 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^35 - 5*x^34 - 120*x^33 + 560*x^32 + 6265*x^31 - 27063*x^30 - 188895*x^29 + 746800*x^28 + 3677845*x^27 - 13130310*x^26 - 48923530*x^25 + 155517810*x^24 + 458129250*x^23 - 1279983515*x^22 - 3063191465*x^21 + 7446901792*x^20 + 14674072475*x^19 - 30859715040*x^18 - 50096318770*x^17 + 91037787910*x^16 + 120155286235*x^15 - 189447334860*x^14 - 197298653000*x^13 + 272612081065*x^12 + 212356717105*x^11 - 261877354307*x^10 - 139209319745*x^9 + 158110983885*x^8 + 48766657725*x^7 - 53789658890*x^6 - 7150843552*x^5 + 8233276160*x^4 + 456884545*x^3 - 363355535*x^2 - 50594420*x - 1782107); 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))];