// Magma code for working with number field 35.35.10738289826578710854795473611086878947834443845310364463749065797704132681.1

// 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 - 2*x^34 - 91*x^33 + 186*x^32 + 3529*x^31 - 7282*x^30 - 77406*x^29 + 160015*x^28 + 1074054*x^27 - 2215841*x^26 - 9975621*x^25 + 20528466*x^24 + 63908089*x^23 - 131541229*x^22 - 286306729*x^21 + 593348019*x^20 + 897545333*x^19 - 1896502635*x^18 - 1942677455*x^17 + 4283320775*x^16 + 2804705170*x^15 - 6752081061*x^14 - 2493142459*x^13 + 7254188151*x^12 + 1058706744*x^11 - 5105442290*x^10 + 147588576*x^9 + 2206890342*x^8 - 365733379*x^7 - 523140235*x^6 + 145322301*x^5 + 51716162*x^4 - 20476030*x^3 - 56585*x^2 + 426253*x - 7523);

// 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 - 2*x^34 - 91*x^33 + 186*x^32 + 3529*x^31 - 7282*x^30 - 77406*x^29 + 160015*x^28 + 1074054*x^27 - 2215841*x^26 - 9975621*x^25 + 20528466*x^24 + 63908089*x^23 - 131541229*x^22 - 286306729*x^21 + 593348019*x^20 + 897545333*x^19 - 1896502635*x^18 - 1942677455*x^17 + 4283320775*x^16 + 2804705170*x^15 - 6752081061*x^14 - 2493142459*x^13 + 7254188151*x^12 + 1058706744*x^11 - 5105442290*x^10 + 147588576*x^9 + 2206890342*x^8 - 365733379*x^7 - 523140235*x^6 + 145322301*x^5 + 51716162*x^4 - 20476030*x^3 - 56585*x^2 + 426253*x - 7523);
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))];