# SageMath code for working with p-adic field 109.1.20.19a1.4.


# Define the p-adic field: 
Prec = 100  # Default precision of 100
Q109 = Qp(109, Prec); x = polygen(QQ)
K.<a> = Q109.extension(x^20 + 11663)

# Defining polynomial: 
K.defining_polynomial()

# Base field Qp: 
K.base()

# Degree over Qp: 
K.absolute_degree()

# Ramification index: 
K.absolute_e()

# Residue field degree (Inertia degree): 
K.absolute_f()

# Maximal unramified subextension: 
K.maximal_unramified_subextension()

# Roots of unity: 
len(K.roots_of_unity())