// Magma code for working with number field 35.35.168109113671617086350535469888345045006842268262463039332288137975520602419531747373726681.1

// Some of these functions may take a long time to execute (this depends on the field).

// Define the number field: 
R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171);

// 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<x> := PolynomialRing(QQ); K<a> := NumberField(x^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171);
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[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];