Finding a prime ideal whose class is order 10 in the ideal class group

Question

"By factorising the ideal $(4+\sqrt{-74})_{R}$, or otherwise, find a prime ideal whose class $[P]$ in the ideal class group $Cl(R)$ has order 10"

$R = \mathbb{Z}[\sqrt{-74}]$

So I've managed to get the prime factorisation: $(4+\sqrt{-74})_R = (2, \sqrt{-74})_R(5, 1-\sqrt{-74})_R(3, 1+\sqrt{-74})_R^2$ but I can't see how leads to a prime ideal whose order in the class group is 10.

I'd love some advice regarding how to proceed

Answer 1

Using Dedekind's theorem it's helpful to first factorise $(2)$, $(3)$ and $ (5) $

We have $(2) = (2,\sqrt(-74))^2 = P_2^2$

$(3) = (3,\sqrt(-74)+1)(3,\sqrt(-74)-1)=P_3Q_3$

$(5) = (5,\sqrt(-74)+1)(5,\sqrt(-74)-1) = P_5Q_5$.

First factorising (75) = $P_3P_5^2$, thus $[Q_3] = [P_5]^2$.

But by the factorisation you have we also get that $(4+\sqrt(-74)) = P_2Q_5P_3^2$

So $[P_2] = [Q_5][P_3]^2$ and so $[P_2] = [Q_5]^5$, $P_2$ has order $ 2$ so $Q_5$ has order $10$