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I am looking for some explicit examples of (elliptic) $K3$-families defined over a number field (better to be over $\mathbb{Q}$) with Picard number $18$ but does not admit Shioda-Inose structure, i.e. there is no primitive embedding from the transcendental lattice to $U^3$ ($U$ is the hyperbolic lattice). I am wondering is there good reference for that?

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    $\begingroup$ Title says Picard number 17; question text says 18. Which do you want? $\endgroup$
    – Noam D. Elkies
    Oct 4, 2018 at 3:57
  • $\begingroup$ @NoamD.Elkies, sorry about that, I want the examples with Picard number 18. $\endgroup$
    – Leo D
    Oct 4, 2018 at 4:49
  • $\begingroup$ OK, got it. If you have a simple (i.e. low-discriminant) example of such a transcendental lattice $T$ then I can write down the surfaces. (If you don't, I can hunt around for $T$ myself but it would take longer.) $\endgroup$
    – Noam D. Elkies
    Oct 4, 2018 at 19:39
  • $\begingroup$ @NoamD.Elkies, thanks. I am sorry, I tried a little while but failed to find a such lattice which can't be primitively embedded into $U^{3}$ and is realizable. $\endgroup$
    – Leo D
    Oct 5, 2018 at 21:37

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