// Magma code for working with number field 38.0.2394510171790650820123124474406353844872595054967993341776661644110188322971389918926750746438903.1

// 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 + 118*x^36 - 435*x^35 + 6131*x^34 - 35235*x^33 + 222008*x^32 - 1286466*x^31 + 5995377*x^30 - 26936649*x^29 + 103513172*x^28 - 328397860*x^27 + 831795745*x^26 - 841346468*x^25 - 6703672958*x^24 + 58117415411*x^23 - 303731983764*x^22 + 1218275129036*x^21 - 3767355799065*x^20 + 7961894667643*x^19 - 2218796696910*x^18 - 81504663492211*x^17 + 502603614584717*x^16 - 2027971305796085*x^15 + 6509260330141933*x^14 - 17551631941921428*x^13 + 40524896388301467*x^12 - 80503012777500047*x^11 + 136388077021556601*x^10 - 193370722349898757*x^9 + 226309207994588218*x^8 - 220960831577107827*x^7 + 187038155057537519*x^6 - 141423416677434751*x^5 + 93202974532409581*x^4 - 51592635735804780*x^3 + 25439606449082092*x^2 - 11768916674310345*x + 3447647463127489);

// 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 + 118*x^36 - 435*x^35 + 6131*x^34 - 35235*x^33 + 222008*x^32 - 1286466*x^31 + 5995377*x^30 - 26936649*x^29 + 103513172*x^28 - 328397860*x^27 + 831795745*x^26 - 841346468*x^25 - 6703672958*x^24 + 58117415411*x^23 - 303731983764*x^22 + 1218275129036*x^21 - 3767355799065*x^20 + 7961894667643*x^19 - 2218796696910*x^18 - 81504663492211*x^17 + 502603614584717*x^16 - 2027971305796085*x^15 + 6509260330141933*x^14 - 17551631941921428*x^13 + 40524896388301467*x^12 - 80503012777500047*x^11 + 136388077021556601*x^10 - 193370722349898757*x^9 + 226309207994588218*x^8 - 220960831577107827*x^7 + 187038155057537519*x^6 - 141423416677434751*x^5 + 93202974532409581*x^4 - 51592635735804780*x^3 + 25439606449082092*x^2 - 11768916674310345*x + 3447647463127489);
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))];