// Magma code for working with number field 35.35.1455622807785591094953547155658149343464416905925495778881639129313848614446169.1 // 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 - 2*x^34 - 121*x^33 + 124*x^32 + 6435*x^31 - 1020*x^30 - 195036*x^29 - 112597*x^28 + 3683986*x^27 + 4452851*x^26 - 44769607*x^25 - 80359740*x^24 + 347984933*x^23 + 850835561*x^22 - 1630509423*x^21 - 5621246939*x^20 + 3615565971*x^19 + 23234081997*x^18 + 3088781711*x^17 - 57619940147*x^16 - 39347873390*x^15 + 76695393961*x^14 + 92864249975*x^13 - 38675667681*x^12 - 94158719636*x^11 - 10380297650*x^10 + 42855721796*x^9 + 16263455858*x^8 - 7226936109*x^7 - 4634581071*x^6 - 47622511*x^5 + 349101928*x^4 + 70736582*x^3 + 5060011*x^2 + 136327*x + 859); // 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 - 2*x^34 - 121*x^33 + 124*x^32 + 6435*x^31 - 1020*x^30 - 195036*x^29 - 112597*x^28 + 3683986*x^27 + 4452851*x^26 - 44769607*x^25 - 80359740*x^24 + 347984933*x^23 + 850835561*x^22 - 1630509423*x^21 - 5621246939*x^20 + 3615565971*x^19 + 23234081997*x^18 + 3088781711*x^17 - 57619940147*x^16 - 39347873390*x^15 + 76695393961*x^14 + 92864249975*x^13 - 38675667681*x^12 - 94158719636*x^11 - 10380297650*x^10 + 42855721796*x^9 + 16263455858*x^8 - 7226936109*x^7 - 4634581071*x^6 - 47622511*x^5 + 349101928*x^4 + 70736582*x^3 + 5060011*x^2 + 136327*x + 859); 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))];