// Magma code for working with number field 46.0.9630272387131177947845223207051634018451530879887040478778044189300817518386752324750061473265804243579.1.
// 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^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

// Discriminant: 
OK := Integers(K); Discriminant(OK);

// Ramified primes: 
PrimeDivisors(Discriminant(OK));

// Automorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Narrow class group: 
NarrowClassGroup(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(Rationals()); K<a> := NumberField(x^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031);
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 pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];