A Square Corona Crystal of the Heronian Triangle 13–14–15: CM Elliptic Curve with 𝑗 = 1728, Torsion Configuration (2,4,4), and a Modular Curve of Level 4

In a previous preprint the author constructed a hexagonal “corona crystal” built from the Heronian triangle 13–14–15 and showed that it sits at a natural node between Pell equations in the real quadratic field Q(√3), a biquadratic field Q(√2,√3), the cyclotomic field Q(ζ12), a wallpaper group of type p6m, and a CM elliptic curve with j-invariant 0 carrying a (2,3,6) torsion configuration, together with a modular interpretation on X₁(6).

In the present preprint we construct and analyze a different square corona crystal built from the same Heronian triangle 13–14–15, now organized around an 8–petal corona and a square translation lattice. We show that:
(i) the resulting wallpaper group is of type p4m, with reflection subgroup isomorphic to the Euclidean Coxeter triangle group Δ(2,4,4);
(ii) after a natural renormalization the translation lattice is the square lattice Z + iZ, so that the crystal lives on the square complex torus T□ = C/(Z + iZ);
(iii) this torus corresponds to a CM elliptic curve with j-invariant 1728, which admits a model E□ : y² = x³ − x with complex multiplication by the Gaussian integers Z[i];
(iv) the square corona crystal induces a natural (2,4,4) torsion configuration on E□ and a distinguished CM point on the modular curve X₁(4) lying above j = 1728.

Together with the hexagonal p6m crystal, the square p4m crystal suggests a broader “CM–crystallographic” program in which Heronian triangles and their rotational coronas are systematically related to CM elliptic curves, torsion configurations of Euclidean triangle type, and modular curves of small level.