# SageMath code for working with number field 20.4.5657954428456562353084024324409254309146809860096000000000000000.2.
# Some of these functions may take a long time to execute (this depends on the field).


# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^20 + 684*x^18 - 120*x^17 + 210872*x^16 - 27840*x^15 + 37744672*x^14 + 2898720*x^13 + 4248246835*x^12 + 2111045760*x^11 + 310035466968*x^10 + 366294835560*x^9 + 14179985825472*x^8 + 26313627836160*x^7 + 288546146778916*x^6 + 569323661850720*x^5 - 4104002620116539*x^4 - 1025151183166560*x^3 - 197619730538519440*x^2 + 186630461902181640*x + 61046467660180710)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Autmorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

# Class group: 
K.class_group().invariants()

# Unit group: 
UK = K.unit_group()

# Unit rank: 
UK.rank()

# Generator for roots of unity: 
UK.torsion_generator()

# Fundamental units: 
UK.fundamental_units()

# Regulator: 
K.regulator()

# Analytic class number formula: 
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ);  K.<a> = NumberField(x^20 + 684*x^18 - 120*x^17 + 210872*x^16 - 27840*x^15 + 37744672*x^14 + 2898720*x^13 + 4248246835*x^12 + 2111045760*x^11 + 310035466968*x^10 + 366294835560*x^9 + 14179985825472*x^8 + 26313627836160*x^7 + 288546146778916*x^6 + 569323661850720*x^5 - 4104002620116539*x^4 - 1025151183166560*x^3 - 197619730538519440*x^2 + 186630461902181640*x + 61046467660180710)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator();  RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))  

# Intermediate fields: 
K.subfields()[1:-1]

# Galois group: 
K.galois_group(type='pari')

# Frobenius cycle types: 
# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]