The binary quartic form $\mathcal{V} = (a, b, c, d, e)$ has the same discriminant $D$ as the binary cubic form \begin{equation}\label{resolvform} \mathcal{C} = \left( 1, - c, b d - 4 a e, 4 a c e - a d^2 - b^2 e \right) . \end{equation} The Hessian $\mathcal{Q}$ of $\mathcal{C}$ and the Jacobian $\mathcal{F}$ of $\mathcal{C}$ have $\mathcal{Q}(1, 0) = i$ and $\mathcal{F}(1, 0) = - j$. Since $\mathcal{C}(1, 0) = 1$, the identity $j^2 + 27 D = 4 i^3$ is a consequence of Cayley's syzygy $\mathcal{F}^2 + 27 D \mathcal{C}^2 = 4 \mathcal{Q}^3$. The forms $\mathcal{V}$ and $\mathcal{C}$ parameterize rings. The matrices \begin{equation*} M_4 = \left( \begin{array}{cccc} u & -a e z & -e (a y+b z) & -e (a x+b y+c z) \\ x & u-b x-c y-d z & -c x-d y-e z & -d x-e y \\ y & a x & u-c y-d z & -d y-e z \\ z & a y & a x+b y & u-d z \\ \end{array} \right) , \end{equation*} and \begin{equation*} M_3 = \left( \begin{array}{ccc} U & -k Y & c k Y-k X \\ X & U+c X-(b d-4 a e) Y & -(b d-4 a e) X-k Y \\ Y & X & U-(b d-4 a e) Y \\ \end{array} \right) , \end{equation*} where $k = 4 a c e - a d^2 - b^2 e$, take care of multiplication in each of these rings via matrix multiplication; the $\alpha = u + x \rho_1 + y \rho_2 + z \rho_3$ can be added and multiplied using matrices. Multiplication tables can be recovered from $M_4$ and $M_3$.

Wood showed that binary quartic forms parameterize rings with monogenic cubic resolvent rings. \begin{align*} A & = \frac{1}{2} \left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -2 \\ \end{array} \right) , & B & = \frac{1}{2} \left( \begin{array}{ccc} 2 e & 0 & d \\ 0 & 2 a & b \\ d & b & 2 c \\ \end{array} \right) , \end{align*} corresponds to the pair of ternary quadratic forms \begin{align*} \mathcal{Q}_1 & = x y - z^2 , & \mathcal{Q}_2 & = e x^2 + a y^2 + b y z + d x z + c z^2 . \end{align*} The pair of matrices $(A, B)$ was then mapped to the binary cubic form \begin{equation*} 4 \det \left( A x + B y \right) = \mathcal{C} . \end{equation*} If I understood correctly, in this way Wood showed that $\mathcal{Q}_1 $ and $\mathcal{Q}_2$ parameterize the same quartic ring that $\mathcal{V}$ does.

Not all quartic rings come from binary quartic forms. For example $y^2 = 4 x^3 - 6075$ has no integral points so the ring of integers of the quartic field of discriminant $225$ is not parameterized by a binary quartic form.

Now let $K = \mathbb{Q} (\zeta )$ be a quartic field of discriminant $\Delta $, where $\zeta $ is a root of $\mathcal{V}(x, 1)$. The discriminant $D$ of $\mathcal{V}$ satisfies $D = \Delta a_0^2 $, where $a_0$ is the index of $\mathbb{Z} [\zeta ]$ in $\mathcal{O}_K$. If $a_0^2$ divides $a$ and $a_0$ divides $b$, then the matrix \begin{equation*} B = \left( \begin{array}{ccc} \frac{3 b^2-8 a c}{a_0^2} & \frac{2 (b c-6 a d)}{a_0} & \frac{b d-16 a e}{a_0} \\ \frac{2 (b c-6 a d)}{a_0} & 4 c^2-8 b d-16 a e & 2 c d-12 b e \\ \frac{b d-16 a e}{a_0} & 2 c d-12 b e & 3 d^2-8 c e \\ \end{array} \right) \end{equation*} is an integral matrix of determinant $16 \Delta $, and hence may be a candidate for one of a pair of matrices $(A, B)$ parameterizing the ring of integers of the quartic number field of discriminant $225$, where $a_0 = 2$ and $(a,b,c,d,e) = (4, -6, 5, -3, 1)$ .

Which pair of ternary quadratic forms in Bhargava's theory parameterize the ring of integers of the quartic number field of discriminant $225$?