# SageMath code for working with number field 44.44.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.1

# 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 - 67*x^42 + 66*x^41 + 2006*x^40 - 1940*x^39 - 35522*x^38 + 33582*x^37 + 415391*x^36 - 381809*x^35 - 3396482*x^34 + 3014673*x^33 + 20088042*x^32 - 17073369*x^31 - 87848856*x^30 + 70775487*x^29 + 288431096*x^28 - 217655609*x^27 - 718686410*x^26 + 501030801*x^25 + 1368730205*x^24 - 867699404*x^23 - 1999332041*x^22 + 1131696716*x^21 + 2238799988*x^20 - 1107666220*x^19 - 1911947202*x^18 + 806431252*x^17 + 1231950206*x^16 - 430200535*x^15 - 588270804*x^14 + 164470157*x^13 + 202516659*x^12 - 43750410*x^11 - 48240456*x^10 + 7822102*x^9 + 7480922*x^8 - 908640*x^7 - 688078*x^6 + 65581*x^5 + 31906*x^4 - 2504*x^3 - 504*x^2 + 24*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 - 67*x^42 + 66*x^41 + 2006*x^40 - 1940*x^39 - 35522*x^38 + 33582*x^37 + 415391*x^36 - 381809*x^35 - 3396482*x^34 + 3014673*x^33 + 20088042*x^32 - 17073369*x^31 - 87848856*x^30 + 70775487*x^29 + 288431096*x^28 - 217655609*x^27 - 718686410*x^26 + 501030801*x^25 + 1368730205*x^24 - 867699404*x^23 - 1999332041*x^22 + 1131696716*x^21 + 2238799988*x^20 - 1107666220*x^19 - 1911947202*x^18 + 806431252*x^17 + 1231950206*x^16 - 430200535*x^15 - 588270804*x^14 + 164470157*x^13 + 202516659*x^12 - 43750410*x^11 - 48240456*x^10 + 7822102*x^9 + 7480922*x^8 - 908640*x^7 - 688078*x^6 + 65581*x^5 + 31906*x^4 - 2504*x^3 - 504*x^2 + 24*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)]