# Oscar code for working with number field 24.4.6769025210045733840131040987091228089411742985248565673828125.5.
# 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 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875)

# 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 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875);
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]