# Oscar code for working with number field 30.30.38731022422755868071217069332926190761458900976553.1

# 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^30 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*x + 1)

# 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^30 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*x + 1);
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]