// Magma code for working with number field 26.26.161976026767757064385262172731473762898927110116516889788623618048.1

// 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^26 - 182*x^24 - 130*x^23 + 13676*x^22 + 16874*x^21 - 557063*x^20 - 898690*x^19 + 13585728*x^18 + 25758668*x^17 - 206672505*x^16 - 437966698*x^15 + 1975279540*x^14 + 4572542860*x^13 - 11606965747*x^12 - 29202022252*x^11 + 39753657307*x^10 + 109898528610*x^9 - 73035512264*x^8 - 227318807846*x^7 + 69544180524*x^6 + 237373608862*x^5 - 38924210832*x^4 - 100654468148*x^3 + 15446207101*x^2 + 2758719782*x + 44790407);

// 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^26 - 182*x^24 - 130*x^23 + 13676*x^22 + 16874*x^21 - 557063*x^20 - 898690*x^19 + 13585728*x^18 + 25758668*x^17 - 206672505*x^16 - 437966698*x^15 + 1975279540*x^14 + 4572542860*x^13 - 11606965747*x^12 - 29202022252*x^11 + 39753657307*x^10 + 109898528610*x^9 - 73035512264*x^8 - 227318807846*x^7 + 69544180524*x^6 + 237373608862*x^5 - 38924210832*x^4 - 100654468148*x^3 + 15446207101*x^2 + 2758719782*x + 44790407);
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))];