// Magma code for working with number field 35.35.705021958507555897735769309192822159832506785420715156309512394727789796888828277587890625.1

// 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 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349);

// 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 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349);
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))];