# Oscar code for working with number field 24.0.22199551505513271594529924008114126848.1.
# 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^24 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41)

# 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))

# Automorphisms: 
automorphism_group(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 = polynomial_ring(QQ); K, a = number_field(x^24 - 4*x^23 + 8*x^22 - 10*x^21 + 70*x^20 - 182*x^19 + 269*x^18 - 372*x^17 + 1647*x^16 - 1870*x^15 + 851*x^14 - 1270*x^13 + 4539*x^12 + 10226*x^11 - 6485*x^10 - 6130*x^9 - 7385*x^8 + 932*x^7 + 26931*x^6 + 14618*x^5 - 4894*x^4 + 334*x^3 + 2048*x^2 - 312*x + 41);
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);
degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

# 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]