# Oscar code for working with number field 42.42.519767234928222794437622861788597020192717533652199079454480438860528408854528.1

# 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^42 - 42*x^40 + 819*x^38 - 9842*x^36 + 81585*x^34 - 494802*x^32 + 2272424*x^30 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7)

# 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^42 - 42*x^40 + 819*x^38 - 9842*x^36 + 81585*x^34 - 494802*x^32 + 2272424*x^30 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7);
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]