# SageMath code for working with number field 44.0.3714575655453538975253519356486345582985254755453847127268314361572265625.1

# 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^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)

# 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^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 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(type='pari')

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