# Oscar code for working with number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.1

# 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^44 - x^43 + 43*x^42 - 39*x^41 + 856*x^40 - 700*x^39 + 10478*x^38 - 7678*x^37 + 88356*x^36 - 57644*x^35 + 545008*x^34 - 314432*x^33 + 2548537*x^32 - 1290809*x^31 + 9238279*x^30 - 4075043*x^29 + 26320891*x^28 - 10020719*x^27 + 59401115*x^26 - 19318239*x^25 + 106508095*x^24 - 29235139*x^23 + 151556799*x^22 - 34552164*x^21 + 170215728*x^20 - 30533255*x^19 + 148629623*x^18 - 11758433*x^17 + 94928616*x^16 + 36112685*x^15 + 30135721*x^14 + 126072207*x^13 - 23633656*x^12 + 220881720*x^11 - 44029406*x^10 + 233441302*x^9 - 35764283*x^8 + 155778230*x^7 - 8938748*x^6 + 18093321*x^5 + 18069001*x^4 + 120313546*x^3 - 192566274*x^2 - 256263936*x + 1026529561)

# 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^44 - x^43 + 43*x^42 - 39*x^41 + 856*x^40 - 700*x^39 + 10478*x^38 - 7678*x^37 + 88356*x^36 - 57644*x^35 + 545008*x^34 - 314432*x^33 + 2548537*x^32 - 1290809*x^31 + 9238279*x^30 - 4075043*x^29 + 26320891*x^28 - 10020719*x^27 + 59401115*x^26 - 19318239*x^25 + 106508095*x^24 - 29235139*x^23 + 151556799*x^22 - 34552164*x^21 + 170215728*x^20 - 30533255*x^19 + 148629623*x^18 - 11758433*x^17 + 94928616*x^16 + 36112685*x^15 + 30135721*x^14 + 126072207*x^13 - 23633656*x^12 + 220881720*x^11 - 44029406*x^10 + 233441302*x^9 - 35764283*x^8 + 155778230*x^7 - 8938748*x^6 + 18093321*x^5 + 18069001*x^4 + 120313546*x^3 - 192566274*x^2 - 256263936*x + 1026529561);
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]