# Oscar code for working with number field 20.10.195532563918689056742899679.1

# 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 compile (this depends on the state of your Julia REPL), and/or to execute (this depends on the field).

# Define the number field: 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 1)

# 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^20 - x^19 - 13*x^18 + 14*x^17 + 70*x^16 - 82*x^15 - 203*x^14 + 261*x^13 + 351*x^12 - 492*x^11 - 397*x^10 + 566*x^9 + 346*x^8 - 403*x^7 - 262*x^6 + 180*x^5 + 142*x^4 - 44*x^3 - 36*x^2 + 1);
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 $p\mathcal{O}_K$ 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]