// Magma code for working with number field 42.42.1236219045653317330764639083558080790009887216550178193020957667462329030021.1 // 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^42 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*x + 1); // 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^42 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*x + 1); 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))];