• Over $\Bbb{Q}^{\mathrm{al}}$, viewing $X$ also as a complex manifold,

$$\begin{aligned} \operatorname{Pic}(X^{\mathrm{al}}) &\simeq H^{1,1}(X_{\Bbb{C}}) \cap H^2(X_{\Bbb{C}}, \Z) \\ &\subset H^2(X_{\Bbb{C}}, \Z) \simeq (-E_8)^2 \oplus U^3 \simeq \Z^{22} \end{aligned}$$

• $\dim H^{1,1}(X_{\Bbb{C}})=20$; $1\leq\rho(X^{\mathrm{al}})\leq20$.
• The degree of "difficulty" is negatively correlated with $\rho(X^{\mathrm{al}})$; $\rho(X^{\mathrm{al}})=1$ is generic.
• $T(X) := \operatorname{Pic}(X^{\mathrm{al}})^{\perp} \subset H^2(X_{\Bbb{C}},\Z)$: the transcendental lattice.
• $T(X)_{\Bbb{Q}}$: minimal rational sub-Hodge structure containing $H^{2,0}(X_{\Bbb{C}})$ after complexification.

  $$H^2(X_{\Bbb{C}},\Bbb{Q}) \simeq \operatorname{Pic}(X^{\mathrm{al}})_{\Bbb{Q}} \oplus T(X)_{\Bbb{Q}}$$

• The "new and interesting" Galois representations arise from $T(X)$.
• In characteristic $p$ the bound $20$ fails: $\rho(X_p^{\mathrm{al}})$ can be as large as $22$.