// Magma code for working with number field 18.18.389573518709461277048874506777900581813141954677569649354154692963827395433931232826260909975966492327717495503836697626432438146795529306112.1.
// 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^18 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312);

// 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(Rationals()); K<a> := NumberField(x^18 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312);
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))];