# SageMath code for working with number field 24.4.6769025210045733840131040987091228089411742985248565673828125.6.
# 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^24 - 2*x^23 + 119*x^22 - 123*x^21 - 2284*x^20 + 35021*x^19 - 287932*x^18 - 3528451*x^17 + 430787*x^16 + 76267881*x^15 + 383091958*x^14 + 1547361734*x^13 + 1926262502*x^12 - 24636617534*x^11 - 181070920997*x^10 - 710772344063*x^9 - 1879981260494*x^8 - 2702189457572*x^7 + 3316968239109*x^6 + 29845357531527*x^5 + 75565793089194*x^4 + 102052357360777*x^3 + 77691878905336*x^2 + 30942834489368*x + 4794247932259)

# 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^24 - 2*x^23 + 119*x^22 - 123*x^21 - 2284*x^20 + 35021*x^19 - 287932*x^18 - 3528451*x^17 + 430787*x^16 + 76267881*x^15 + 383091958*x^14 + 1547361734*x^13 + 1926262502*x^12 - 24636617534*x^11 - 181070920997*x^10 - 710772344063*x^9 - 1879981260494*x^8 - 2702189457572*x^7 + 3316968239109*x^6 + 29845357531527*x^5 + 75565793089194*x^4 + 102052357360777*x^3 + 77691878905336*x^2 + 30942834489368*x + 4794247932259)
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)]