Formula for the jinvariant of a ternary cubic
The jinvariant of a ternary cubic

Maple computation and Macaulay2 computation for Lemma 3
We compute the ideal describing the condition that a linear form divides a ternary cubic. The maple code is used to produce the input to Macaulay2. Specializing this ideal to the case that the cubic is in the Hesse pencil of a given cubic f gives a Grobner basis whose triangular form shows that the inflection points lie in a solvable extension. The full output is here. 
Mathematica computation of the inflection points of a general smooth cubic
This Mathematica code carries out the computation described in Section 2 to compute an explicit formula, in radicals, for the nine inflection points of a general smooth cubic. The full formula, which occupies 33,013 KB, can be found here. Note that the invariants S and T are left unexpanded in terms of the ten coefficients; otherwise, the formula would be many gigabytes long. 
Computation of a 3x3 matrix taking a given cubic to honeycomb form (forthcoming)

Mathematica algorithm to compute honeycomb embeddings
We illustrate our Theorem 7 by producing, for a given jinvariant and a given choice of nine nonzero field elements p1,...,p9 satisfying the conditions of Section 3, a parametrization by theta functions of the associated honeycomb embedding, and the coefficients cijk which give the implicit representation of our honeycomb elliptic curve. Our code is demonstrated on a sample choice of nine points given in Section 4. 