# Oscar code for working with number field 18.6.51624177137754227336536682168566437749832086648167693412837925000000000000.1.
# 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 execute (this depends on the field).


# Define the number field: 
Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019)

# 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^18 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019);
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 pO_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]