// Magma code for working with number field 38.0.2501696311112367702213593384284049957523786814936027691413131289153060507631187229631.1

// 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 + 3*x^36 - 11*x^35 + 44*x^34 + 732*x^33 - 116*x^32 + 2069*x^31 + 2818*x^30 - 12880*x^29 + 93246*x^28 - 145805*x^27 + 92780*x^26 + 2105124*x^25 - 3183002*x^24 - 3104601*x^23 + 12583923*x^22 + 9706311*x^21 + 21916307*x^20 - 49619666*x^19 + 22159929*x^18 - 196911474*x^17 + 1026982112*x^16 + 311788273*x^15 - 1106984612*x^14 - 334730951*x^13 - 1752540110*x^12 + 3801710744*x^11 + 7280378790*x^10 + 1308968234*x^9 - 6926602921*x^8 - 6856588968*x^7 + 18604485712*x^6 + 18058309277*x^5 + 9272912074*x^4 - 5214869030*x^3 - 1463327624*x^2 + 5326526527*x + 2048986499);

// 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 + 3*x^36 - 11*x^35 + 44*x^34 + 732*x^33 - 116*x^32 + 2069*x^31 + 2818*x^30 - 12880*x^29 + 93246*x^28 - 145805*x^27 + 92780*x^26 + 2105124*x^25 - 3183002*x^24 - 3104601*x^23 + 12583923*x^22 + 9706311*x^21 + 21916307*x^20 - 49619666*x^19 + 22159929*x^18 - 196911474*x^17 + 1026982112*x^16 + 311788273*x^15 - 1106984612*x^14 - 334730951*x^13 - 1752540110*x^12 + 3801710744*x^11 + 7280378790*x^10 + 1308968234*x^9 - 6926602921*x^8 - 6856588968*x^7 + 18604485712*x^6 + 18058309277*x^5 + 9272912074*x^4 - 5214869030*x^3 - 1463327624*x^2 + 5326526527*x + 2048986499);
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))];