# SageMath code for working with number field 44.0.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.3

# 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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625)

# 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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625)
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)]