1. We need to find the number of 18-tuples $(a_0, a_1, \dots, a_{17})$ where each $a_i \in \{-1, 0, 1\}$ and they satisfy the equation:
$$a_0 + 2a_1 + 2^2a_2 + \cdots + 2^{17}a_{17} = 2^{10}$$

2. Notice that $2^{10}$ is a power of 2, and we need to express it as a sum of terms of the form $2^i a_i$ where $a_i \in \{-1, 0, 1\}$.

3. We can write $2^{10}$ as a sum of distinct powers of 2, considering the constraints on $a_i$. The possible ways to write $2^{10}$ using the terms $2^i a_i$ are:
$$2^{10}, \quad 2^{11} - 2^{10}, \quad 2^{12} - 2^{11} - 2^{10}, \quad \ldots, \quad 2^{17} - 2^{16} - \cdots - 2^{11} - 2^{10}$$

4. Each of these expressions represents a unique way to form $2^{10}$ using the terms $2^i a_i$ with $a_i \in \{-1, 0, 1\}$.

5. To count the number of such sets, we observe that each expression corresponds to a unique combination of $a_i$ values. Specifically, each expression involves a different set of indices $i$ where $a_i = 1$ and the rest are 0 or -1.

6. The number of such distinct sets is equal to the number of terms in the sequence from $2^{10}$ to $2^{17} - 2^{16} - \cdots - 2^{11} - 2^{10}$. This sequence has 8 terms.

Conclusion:
$$\boxed{8}$$