// Magma code for working with number field 38.0.2233638411813024816853081773648251688534529753590642239923912316757382599022775822751448518259.1

// 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 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023);

// 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 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023);
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))];