Elliptic modular invariants of 4-by-4 matrices

Let A be an n-by -n matrix. The ternary form FA(t, x, y) = det(tIn + xR(A) + y(sic)(A)) characterizes the numerical range of A. For n = 4, we assume that the ternary form FA(t, x, y) is irreducible and the complex projective algebraic curve FA(t, x, y) = 0 is elliptic. We prove that the number of analytic curves composing the real algebraic curve of F-A(t, x, y) = 0 is 2. This result is applied to show that the j-invariant of the elliptic curve F-A(t, x, y) = 0 is greater than or equal to 1. (c) 2023 Elsevier Inc. All rights reserved.