Discriminant Loci, Galois Symmetry, and Cm Specializations in a Cubic Family Arising From an Arc–difference Elimination Problem

A symbolic elimination problem motivated by an arc--difference system produces (i) a mixed-derivative identity in four variables and (ii) a degree-12 univariate elimination polynomial. We isolate the algebraic core of these outputs and interpret the resulting arithmetic geometry. First, we show that the mixed-derivative identity is equivalent, after a dimensionless normalization, to a one-parameter cubic equationₜ (z) =t z³-3t z²+ (3t+2) z+ (1-t) =0 the variable \ (z= (q-s) ²/w²\). The six radical branches observed in computer algebra output correspond to the three roots of \ (Pₜ\) together with the independent sign choice \ (q=s wz\). We compute\ᵦ (Pₜ) =-t (243t+32) parametrize the locus on which the discriminant is a square, yielding a rational conic. Second, we attach the genus-1 curve \ (Eₜ: y²=Pₜ (z) \) and obtain a short Weierstrass modelₜ: \ y²=t x³+2x+3, (Eₜ) =5529632+243t. particular, under the natural positivity conditions inherited from the original system (forcing \ (t>0\) ), one has \ (00}\). For completeness we also record the finite set of rational specializations \ (t\) for which \ (Eₜ\) has CM (equivalently, for which \ (j (Eₜ) \) is a rational CM \ (j\) -invariant). Third, we embed \ (Pₜ\) into a two-parameter deformation \ (Pₓ, ₊\) and identify the constant-CM subfamilies \ (k=0\) (constant \ (j=0\) ) and \ (k=-1\) (constant \ (j=1728\) ). Subsequent sections analyze the degree-12 elimination family, including a balanced specialization yielding a regular \ (S₁₁\) -extension over \ ( () \), together with reproducible computations supplied as supplementary SageMath material.