// Magma code for working with number field 35.35.73261800077965937220382205398471606200231960977600836588587975360331959691780081.1

// 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^35 - 7*x^34 - 112*x^33 + 742*x^32 + 5845*x^31 - 34447*x^30 - 188041*x^29 + 918544*x^28 + 4120599*x^27 - 15488536*x^26 - 63930916*x^25 + 170889838*x^24 + 709660070*x^23 - 1224719069*x^22 - 5607885533*x^21 + 5331715046*x^20 + 31035849173*x^19 - 10542390278*x^18 - 116936498434*x^17 - 15812866636*x^16 + 286852619543*x^15 + 146466192098*x^14 - 426702558436*x^13 - 357822103121*x^12 + 338413935629*x^11 + 421000631443*x^10 - 94811511375*x^9 - 240823440823*x^8 - 25297046987*x^7 + 58988457696*x^6 + 13678049308*x^5 - 6371815744*x^4 - 1687698845*x^3 + 280068103*x^2 + 55092156*x - 2660503);

// 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^35 - 7*x^34 - 112*x^33 + 742*x^32 + 5845*x^31 - 34447*x^30 - 188041*x^29 + 918544*x^28 + 4120599*x^27 - 15488536*x^26 - 63930916*x^25 + 170889838*x^24 + 709660070*x^23 - 1224719069*x^22 - 5607885533*x^21 + 5331715046*x^20 + 31035849173*x^19 - 10542390278*x^18 - 116936498434*x^17 - 15812866636*x^16 + 286852619543*x^15 + 146466192098*x^14 - 426702558436*x^13 - 357822103121*x^12 + 338413935629*x^11 + 421000631443*x^10 - 94811511375*x^9 - 240823440823*x^8 - 25297046987*x^7 + 58988457696*x^6 + 13678049308*x^5 - 6371815744*x^4 - 1687698845*x^3 + 280068103*x^2 + 55092156*x - 2660503);
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))];