# SageMath code for working with p-adic field 7.3.7.21a14.1.


# Define the p-adic field: 
Prec = 100  # Default precision of 100
Q7 = Qp(7, Prec); x = polygen(QQ)
L.<t> = Q7.extension(x^3 + 6*x^2 + 4)
K.<a> = L.extension(x^7 + (35*t + 7)*x + 7)

# 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())