# Oscar code for working with number field 18.18.389573518709461277048874506777900581813141954677569649354154692963827395433931232826260909975966492327717495503836697626432438146795529306112.1.
# If you have not already loaded the Oscar package, you should type "using Oscar;" before running the code below.
# Some of these functions may take a long time to execute (this depends on the field).


# Define the number field: 
Qx, x = polynomial_ring(QQ); K, a = number_field(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: 
defining_polynomial(K)

# Degree over Q: 
degree(K)

# Signature: 
signature(K)

# Discriminant: 
OK = ring_of_integers(K); discriminant(OK)

# Ramified primes: 
prime_divisors(discriminant((OK)))

# Autmorphisms: 
automorphisms(K)

# Integral basis: 
basis(OK)

# Class group: 
class_group(K)

# Unit group: 
UK, fUK = unit_group(OK)

# Unit rank: 
rank(UK)

# Generator for roots of unity: 
torsion_units_generator(OK)

# Fundamental units: 
[K(fUK(a)) for a in gens(UK)]

# Regulator: 
regulator(K)

# Analytic class number formula: 
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); 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 = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

# Intermediate fields: 
subfields(K)[2:end-1]

# Galois group: 
G, Gtx = galois_group(K); G, transitive_group_identification(G)

# Frobenius cycle types: 
# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]