Is the ideal $\displaystyle (1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})$ maximal in$\displaystyle \mathbb{Z}[\sqrt{2}]$?

I find that $\displaystyle gcd(1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})=(-1-\sqrt{2})$, which is irreducible in $\displaystyle \mathbb{Z}[\sqrt{2}].

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Does this mean that the ideal is maximal?

I guess the question is equivalent to the question is $\displaystyle \mathbb{Z}[\sqrt{2}]$ Euclidean doman, but I don't know is it.

Is there a rule when $\displaystyle \mathbb{Z}[d]$ is Euclidean domain, depending on $\displaystyle d$?

Thank you.