// Magma code for working with number field 23.23.39941131008096550428701851231676209894223228004485139129721.1 // 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^23 - x^22 - 220*x^21 + 87*x^20 + 19764*x^19 + 7502*x^18 - 942100*x^17 - 1247222*x^16 + 25503060*x^15 + 62076437*x^14 - 368245290*x^13 - 1464919087*x^12 + 1795768610*x^11 + 16428532486*x^10 + 16509241674*x^9 - 60887740509*x^8 - 181108379413*x^7 - 148388500129*x^6 + 88587002490*x^5 + 252590462192*x^4 + 171337144724*x^3 + 31220491402*x^2 - 10221385976*x - 3012210881); // 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^23 - x^22 - 220*x^21 + 87*x^20 + 19764*x^19 + 7502*x^18 - 942100*x^17 - 1247222*x^16 + 25503060*x^15 + 62076437*x^14 - 368245290*x^13 - 1464919087*x^12 + 1795768610*x^11 + 16428532486*x^10 + 16509241674*x^9 - 60887740509*x^8 - 181108379413*x^7 - 148388500129*x^6 + 88587002490*x^5 + 252590462192*x^4 + 171337144724*x^3 + 31220491402*x^2 - 10221385976*x - 3012210881); 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))];