// Magma code for working with number field 38.0.10456376296029042882633731329285387968875960938724861754483238620568507960573755104483627713707.1

// 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^38 - x^37 + 109*x^36 - 318*x^35 + 7607*x^34 - 26082*x^33 + 346271*x^32 - 1277601*x^31 + 11625429*x^30 - 40847577*x^29 + 278354747*x^28 - 890102893*x^27 + 4876149460*x^26 - 13377962627*x^25 + 59213347219*x^24 - 129738898417*x^23 + 497734200105*x^22 - 864588650146*x^21 + 3071362959747*x^20 - 3938571660134*x^19 + 13635086341776*x^18 - 11900678948488*x^17 + 45427254629183*x^16 - 23117165748554*x^15 + 104447793269632*x^14 - 20670309199584*x^13 + 172563580620384*x^12 + 14666610274324*x^11 + 167592119930922*x^10 + 69948796872392*x^9 + 125341715722354*x^8 + 63681900889938*x^7 + 64136735752211*x^6 + 36520042111943*x^5 + 22298249070088*x^4 + 8329262399475*x^3 + 2645555104756*x^2 + 468857415513*x + 63175314409);

// 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^38 - x^37 + 109*x^36 - 318*x^35 + 7607*x^34 - 26082*x^33 + 346271*x^32 - 1277601*x^31 + 11625429*x^30 - 40847577*x^29 + 278354747*x^28 - 890102893*x^27 + 4876149460*x^26 - 13377962627*x^25 + 59213347219*x^24 - 129738898417*x^23 + 497734200105*x^22 - 864588650146*x^21 + 3071362959747*x^20 - 3938571660134*x^19 + 13635086341776*x^18 - 11900678948488*x^17 + 45427254629183*x^16 - 23117165748554*x^15 + 104447793269632*x^14 - 20670309199584*x^13 + 172563580620384*x^12 + 14666610274324*x^11 + 167592119930922*x^10 + 69948796872392*x^9 + 125341715722354*x^8 + 63681900889938*x^7 + 64136735752211*x^6 + 36520042111943*x^5 + 22298249070088*x^4 + 8329262399475*x^3 + 2645555104756*x^2 + 468857415513*x + 63175314409);
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))];