Computing the Picard group of $\mathbb{Z}[\sqrt{-19}]$

Question

I want to compute the Picard group of $\mathbb{Z}[\sqrt{-19}]$, which is not a Dedekind domain. The problem is that I don't even know where to begin.

Any ideas would be helpful.

Thanks

Edit- Since there's some confusion by what I mean by the "Picard group".

$\textit{Definition}$

Let A be an integral domain. Let K denote it's field of fractions. Consider the set of all invertible fractional ideals F(A). Then F(A) forms a group. Consider the subgroup of non-zero principal fractional ideals P(A). So any element in P(A) looks like xA where x $\in$ K$^*$. Then the Picard group of A is defined as F(A)/P(A).

When A is a Dedekind domain, the Picard group coincides with the class group.

Edit 2 - So here's why I'm having a hard time with this problem.

Had $\mathbb{Z}\sqrt{-19}$ been integrally closed, we could proceed using the Minkowski bound and Kummer-Dedekind theory. Unfortunately, $\mathbb{Z}[\sqrt{-19}]$ is $\textit{not}$ integrally closed.

The integral closure in it's field of fractions $\mathbb{Q}(\sqrt{-19})$ is actually strictly bigger, given by $\mathbb{Z}[\frac{1 + \sqrt{-19}}{2}]$.

And thus, I can't use routine methods to try and understand this group.

Answer 1

You want to use the Kummer-Dedekind Theorem to factor the prime ideals of norm at most the Minkowski bound.

I recommend you follow 2.6.1 of https://jrsijsling.eu/notes/ant-notes.pdf

Answer 2

In case of imaginary quadratic field, there is a nice correspondence between the Picard group of orders with discriminant $D$ and reduced primitive quadratic form with the same discriminant. More precisely, the form $ax^2+bxy+cy^2$ corresponds to the ideal class of $(a,(-b+\sqrt{D})/2)$.

For more information and proof, consult chapter 7 of the well-known Primes of the Form $x^2+ny^2$ by David Cox.

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The order $\mathbb{Z}[\sqrt{-19}]$ has discriminant $-56$, and there are three primitive reduced forms: $$x^2+19y^2 \qquad 4x^2-2xy+5y^2 \qquad 4x^2+2xy+5y^2$$ they respectively translate to the trivial ideal, $(4,1+\sqrt{-19})$ and $(4,-1+\sqrt{-19})$.

Hence the Picard group of $\mathbb{Z}[\sqrt{-19}]$ is cyclic of order $3$ generated by $(4,1+\sqrt{-19})$.