# SageMath code for working with number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.3

# 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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416)

# 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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416)
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)]