# Oscar code for working with number field 35.35.10738289826578710854795473611086878947834443845310364463749065797704132681.1

# 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^35 - 2*x^34 - 91*x^33 + 186*x^32 + 3529*x^31 - 7282*x^30 - 77406*x^29 + 160015*x^28 + 1074054*x^27 - 2215841*x^26 - 9975621*x^25 + 20528466*x^24 + 63908089*x^23 - 131541229*x^22 - 286306729*x^21 + 593348019*x^20 + 897545333*x^19 - 1896502635*x^18 - 1942677455*x^17 + 4283320775*x^16 + 2804705170*x^15 - 6752081061*x^14 - 2493142459*x^13 + 7254188151*x^12 + 1058706744*x^11 - 5105442290*x^10 + 147588576*x^9 + 2206890342*x^8 - 365733379*x^7 - 523140235*x^6 + 145322301*x^5 + 51716162*x^4 - 20476030*x^3 - 56585*x^2 + 426253*x - 7523)

# 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^35 - 2*x^34 - 91*x^33 + 186*x^32 + 3529*x^31 - 7282*x^30 - 77406*x^29 + 160015*x^28 + 1074054*x^27 - 2215841*x^26 - 9975621*x^25 + 20528466*x^24 + 63908089*x^23 - 131541229*x^22 - 286306729*x^21 + 593348019*x^20 + 897545333*x^19 - 1896502635*x^18 - 1942677455*x^17 + 4283320775*x^16 + 2804705170*x^15 - 6752081061*x^14 - 2493142459*x^13 + 7254188151*x^12 + 1058706744*x^11 - 5105442290*x^10 + 147588576*x^9 + 2206890342*x^8 - 365733379*x^7 - 523140235*x^6 + 145322301*x^5 + 51716162*x^4 - 20476030*x^3 - 56585*x^2 + 426253*x - 7523);
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]