# SageMath code for working with number field 44.0.6819629148478549071885414510662846662953220513498055755117351820034888330697796222976.1

# 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 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304)

# 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 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304)
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)]