# Oscar code for working with number field 40.0.8600477427850631837563790410386763343300628153464218251684553761.1

# 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^40 - x^39 + 2*x^38 - 5*x^37 + 5*x^36 - 10*x^35 + 17*x^34 - 17*x^33 + 34*x^32 - 45*x^31 + 45*x^30 - 23*x^29 + 22*x^28 + 45*x^27 - 157*x^26 + 250*x^25 - 585*x^24 + 969*x^23 - 1426*x^22 + 2565*x^21 - 3589*x^20 + 5130*x^19 - 5704*x^18 + 7752*x^17 - 9360*x^16 + 8000*x^15 - 10048*x^14 + 5760*x^13 + 5632*x^12 - 11776*x^11 + 46080*x^10 - 92160*x^9 + 139264*x^8 - 139264*x^7 + 278528*x^6 - 327680*x^5 + 327680*x^4 - 655360*x^3 + 524288*x^2 - 524288*x + 1048576)

# 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^40 - x^39 + 2*x^38 - 5*x^37 + 5*x^36 - 10*x^35 + 17*x^34 - 17*x^33 + 34*x^32 - 45*x^31 + 45*x^30 - 23*x^29 + 22*x^28 + 45*x^27 - 157*x^26 + 250*x^25 - 585*x^24 + 969*x^23 - 1426*x^22 + 2565*x^21 - 3589*x^20 + 5130*x^19 - 5704*x^18 + 7752*x^17 - 9360*x^16 + 8000*x^15 - 10048*x^14 + 5760*x^13 + 5632*x^12 - 11776*x^11 + 46080*x^10 - 92160*x^9 + 139264*x^8 - 139264*x^7 + 278528*x^6 - 327680*x^5 + 327680*x^4 - 655360*x^3 + 524288*x^2 - 524288*x + 1048576);
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]