# SageMath code for working with number field 40.0.31654584865659568778929513407372752241664000000000000000000000000000000.1.
# 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^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

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

# Automorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

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

# Narrow class group: 
K.narrow_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^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1)
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()

# 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)]