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The class group $\operatorname{Cl}(\mathcal{O}_K)$ has order $4$ and consists of $ \langle 1 \rangle, \langle 2, 1+\sqrt{-17} \rangle, \langle 3, 1 + \sqrt{-17} \rangle$ and $\langle 3, 2 + \sqrt{-17} \rangle$.

One elliptic curve with CM by $\mathbb{Z}[\sqrt{-17}]$ is just $\mathbb{C}/\mathbb{Z}[\sqrt{-17}]$. From the action of $\operatorname{Cl}(\mathcal{O}_K)$ on elliptic curves with CM by $\mathcal{O}_K$ we get three more elliptic curves, given by lattices homothetic to $\mathbb{Z}\big[\frac{1+\sqrt{-17}}{2}\big]$, $\mathbb{Z}\big[\frac{1+\sqrt{-17}}{3}\big]$, and $\mathbb{Z}\big[\frac{2+\sqrt{-17}}{3}\big]$.

The $j$-invariant of $\mathbb{C}/\mathbb{Z}[\sqrt{-17}]$ is $j(\sqrt{-17})$ which is an algebraic integer of degree $\#\operatorname{Cl}(\mathcal{O}_K) = 4$ whose conjugates are exactly [notes, Theorem 11] the $j$-invariants of the elliptic curves above, i.e. $j\big(\frac{1+\sqrt{-17}}{2}\big), j\big(\frac{1+\sqrt{-17}}{3}\big)$ and $j\big(\frac{2+\sqrt{-17}}{3}\big)$.

The approximate values of these numbers can be found using PARI/GP's ellj :

? ellj(sqrt(-17)) %1 = 178211465582.57236317285989152242246715 ? ellj((1+sqrt(-17))/2) %2 = -421407.46393796828425027276471142244105 ? ellj((1+sqrt(-17))/3) %3 = -2087.5542126022878206248288778733953807 - 4843.8060029534518331212466414598790536*I ? ellj((2+sqrt(-17))/3) %4 = -2087.5542126022878206248288778733953807 + 4843.8060029534518331212466414598790536*I

Now $j(\sqrt{-17})$ is a root of $f = \prod (x-j_k) \in \mathbb{Z}[x]$ where the $j_k$ are all the conjugates. We can approximate $f$ by replacing the $j_k$ with their numerical approximations and expanding the product. In this way we find that $j(\sqrt{-17})$ is very probably the unique positive real root of $$x^4 - 178211040000x^3 - 75843692160000000x^2 - 318507038720000000000x - 2089297506304000000000000,$$ namely $$8000 (5569095 + 1350704 \sqrt{17} + 4 \sqrt{3876889241278 + 940283755330 \sqrt{17}}),$$ which can be verified numerically to very high precision.

As mentioned in the comments we can also find an explicit Weierstrass form for an elliptic curve with given $j$-invariant.