as a project for a class I'm currently taking, I've decided to undertake the execution of the proofs in Manjul Bhargava's expositional article, "Higher composition laws I: A new view on Gauss composition, and quadratic generalizations", which I've linked to here: https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n1-p03.pdf
I'm having trouble with the proofs of some of his main theorems, which involve solving systems of 18, 94, and 135 linear and quadratic equations, respectively. I would appreciate any advice regarding how one might carry out these computations using a computer algebra system. I've provided below the basic setup for the case with 135 equations (directly copied from the article). We have: $$\det(\alpha_i,\alpha_j,\alpha_k)= c_{ijk}+a_{ijk}\tau$$where the 40 integers $\{a_{ijk}\}$ and $\{c_{ijk}\}$ satisfy $$c_{ijk} = -c_{jik} = -c_{ikj} = -c_{kji}$$and$$a_{ijk} = -a_{jik} = -a_{ikj} = -a_{kji}$$respectively (in the above, $i,j,k\in\{1,2,3,4,5,6\}$). We also have the identity: 
for all $i,j,k,l,m,n\in\{1,2,3,4,5,6\}$ (The 35 is the equation number, not part of the equation). We also have $$\tau^2 = \frac{D-\epsilon}{4}+\epsilon\tau$$where $\epsilon=D\mod 4$, and $D$ is a fixed integer (the discriminant of the quadratic order we're working with) that is either 0 or 1 modulo 4 (i.e. we only need consider the cases $\epsilon =0$ and $\epsilon =1$).
I have several problems plaguing me in trying to approach these equations. First, I'm unsure how to produce the 135 (or the 94, or the 18) linear and quadratic equations (that we get from expanding and equating the powers of 1 and $\tau$ in the resulting equations) using a computer algebra system. I can see how these equations could be produced by hand, but am not sure how to reproduce this on a computer. Is there some way I could get the equations to be the generators of an ideal in Macaulay2 without having to enter them manually?. Second, I don't know how one might go about solving for the $c_{ijk}$ in terms of the $a_{ijk}$. Thank you!