# SageMath code for working with number field 24.4.35863302598527284385359953566442393477815766818821430206298828125.2.
# 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^24 - 8*x^23 + 64*x^22 - 902*x^21 + 6316*x^20 - 136934*x^19 + 609372*x^18 - 8671906*x^17 + 30971223*x^16 - 528120304*x^15 + 2227739352*x^14 - 27368555416*x^13 + 84384989853*x^12 - 980595596854*x^11 + 3158183643447*x^10 - 24849991956160*x^9 + 74624141823520*x^8 - 442422095928880*x^7 + 765727826720400*x^6 - 2316315944560320*x^5 - 3888629115012928*x^4 - 1094970506171776*x^3 - 29289495523457792*x^2 - 50029902292521984*x - 21580162302394368)

# 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 - 8*x^23 + 64*x^22 - 902*x^21 + 6316*x^20 - 136934*x^19 + 609372*x^18 - 8671906*x^17 + 30971223*x^16 - 528120304*x^15 + 2227739352*x^14 - 27368555416*x^13 + 84384989853*x^12 - 980595596854*x^11 + 3158183643447*x^10 - 24849991956160*x^9 + 74624141823520*x^8 - 442422095928880*x^7 + 765727826720400*x^6 - 2316315944560320*x^5 - 3888629115012928*x^4 - 1094970506171776*x^3 - 29289495523457792*x^2 - 50029902292521984*x - 21580162302394368)
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)]