// Magma code for working with p-adic field 5.2.10.20a3.2.


// Define the p-adic field: 
Prec := 100;  // Default precision of 100
Q5 := pAdicField(5, Prec);
K := LocalField(Q5, Polynomial(Q5, [1029, 20490, 189460, 1075205, 4189440, 11882496, 25390080, 41748480, 53527680, 53941760, 42904960, 26970880, 13381920, 5218560, 1586880, 371328, 65460, 8400, 740, 40, 1]));

// Defining polynomial: 
DefiningPolynomial(K);

// Base field Qp: 
Q5;

// Degree over Qp: 
Degree(K);

// Ramification index: 
RamificationIndex(K);

// Residue field degree (Inertia degree): 
InertiaDegree(K);

// Discriminant exponent: 
Valuation(Discriminant(K));