// Magma code for working with number field 44.0.6819629148478549071885414510662846662953220513498055755117351820034888330697796222976.1 // 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 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304); // 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 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304); 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))];