[Here](https://mathoverflow.net/a/55808/138661) the following is stated:



> 
> It's a basic fact in $p$-adic Hodge theory that any 2-dim. absolutely
>  irreducible $G\_{\mathbb Q\_p}$-representation with distinct Hodge-Tate weights is uniquely determined by $a\_p$.
> 
> 
> 


Could somebody explain in maximum detail why is this true? Taking inspiration from the famous ELI5 communication style, explain like I am a generic first-year grad student.



2-dimensional absolutely irreducible $p$-adic Galois reps
See [these](http://math.stanford.edu/~conrad/papers/notes.pdf) notes on Hodge theory, Th. 8.3.6.


