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We determine reductions of 2-dimensional, irreducible, semi-stable, andnon-crystalline representations of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbbQ_p)$ with Hodge--Tate weights $0 < k-1$ and with $\mathcal L$-invariant whose$p$-adic norm is sufficiently large, depending on $k$. Our main result providesthe first systematic examples of the reductions for $k \geq p$.

Reductions of $2$-dimensional semi-stable representations with large $\mathcal L$-invariant