// Magma code for working with number field 34.34.342523005011894297428856269332610453116457630461733441736562419892654124149.1

// 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^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967);

// 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^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967);
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))];