// Magma code for working with number field 40.0.3217724369317177596907031731223092219803204403094256869508157994902742845244016168900761.1

// 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^40 - x^39 + 165*x^38 - 165*x^37 + 12629*x^36 - 12629*x^35 + 595157*x^34 - 595157*x^33 + 19330517*x^32 - 19330517*x^31 + 458986965*x^30 - 458986965*x^29 + 8247186901*x^28 - 8247186901*x^27 + 114533209557*x^26 - 114533209557*x^25 + 1245029995989*x^24 - 1245029995989*x^23 + 10665836549589*x^22 - 10665836549589*x^21 + 72174586435029*x^20 - 72174586435029*x^19 + 385310040397269*x^18 - 385310040397269*x^17 + 1616256307697109*x^16 - 1616256307697109*x^15 + 5295568227538389*x^14 - 5295568227538389*x^13 + 13471816938296789*x^12 - 13471816938296789*x^11 + 26553814875510229*x^10 - 26553814875510229*x^9 + 40944012606445013*x^8 - 40944012606445013*x^7 + 51101799240046037*x^6 - 51101799240046037*x^5 + 55223799613101525*x^4 - 55223799613101525*x^3 + 56012699206030805*x^2 - 56012699206030805*x + 56057779182769621);

// 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^40 - x^39 + 165*x^38 - 165*x^37 + 12629*x^36 - 12629*x^35 + 595157*x^34 - 595157*x^33 + 19330517*x^32 - 19330517*x^31 + 458986965*x^30 - 458986965*x^29 + 8247186901*x^28 - 8247186901*x^27 + 114533209557*x^26 - 114533209557*x^25 + 1245029995989*x^24 - 1245029995989*x^23 + 10665836549589*x^22 - 10665836549589*x^21 + 72174586435029*x^20 - 72174586435029*x^19 + 385310040397269*x^18 - 385310040397269*x^17 + 1616256307697109*x^16 - 1616256307697109*x^15 + 5295568227538389*x^14 - 5295568227538389*x^13 + 13471816938296789*x^12 - 13471816938296789*x^11 + 26553814875510229*x^10 - 26553814875510229*x^9 + 40944012606445013*x^8 - 40944012606445013*x^7 + 51101799240046037*x^6 - 51101799240046037*x^5 + 55223799613101525*x^4 - 55223799613101525*x^3 + 56012699206030805*x^2 - 56012699206030805*x + 56057779182769621);
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))];