... values¹

Actually, we have to discard those values where $\Delta = -4b^3-27c^2 = 0$ since these correspond to singular curves

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... general²

You can find histograms for all primes $5\leq p\leq 293$ at http://www.math.utah.edu/$\sim$jfernand/teaching/elliptic/docs/histograms.pdf

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... square³

This follows from the fact that, since $k$ is a non-square so is $k^{-1}$ and the product of two non-squares in $\ensuremath{\mathbb{Z}}_p$ is a square (exercise).

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... extension⁴

A field extension is a field containing the given field as a subfield. For example, $\ensuremath{\mathbb{R}}$ is a field extension of $\ensuremath{\mathbb{Q}}$ and $\ensuremath{\mathbb{C}}$ is a field extension of both. A field extension of $\ensuremath{\mathbb{Z}}_5$ is, for instance, the field $\ensuremath{\mathbb{Z}}_5[\sqrt{2}] = \{a+b\sqrt{2}:a,b\in \ensuremath{\mathbb{Z}}_5\}$.

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... class⁵

Still, one may wonder, all these curves are isomorphic in some extension field. What is the extension required? A short computation shows that an extension of order $4$ is required. For example, in the context of Example 3, $y^2=x^3+x$ and $y^2=x^3+3x$ are isomorphic over $\ensuremath{\mathbb{Z}}_5[\sqrt[4]{3}]$.

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... valid⁶

In this case we may need to consider extensions of order $6$ of $\ensuremath{\mathbb{Z}}_p$ to realize the isomorphism.

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