Elliptic Curves Associated to the Pisot-Boundary Polynomials x³ − x − 1 and x⁴ − x − 1

The polynomials x³ − x − 1 and x⁴ − x − 1 define elliptic curves over ℚ: Cremona 368f1 (LMFDB 368.g1) and Cremona 1132b1 (LMFDB 1132.c1). We catalog their arithmetic invariants — conductors, discriminants, ranks, Mordell–Weil groups, L-functions, modular forms, and Tamagawa numbers — and identify structural parallels: both have negative prime number field discriminants (−23, −283), class number 1, trivial torsion and Tate–Shafarevich groups, and period ratios on the boundary of the fundamental domain. The rank asymmetry (0 vs. 1) maps the Pisot boundary between n = 3 and n = 4. A graded exclusion test over 1,676 polynomial families xⁿ + ax + b confirms that these arithmetic conditions jointly select a unique pair. All claims are verifiable in minutes using PARI/GP; commands are provided in an appendix.Verification notebook: https://colab.research.google.com/drive/1nTxRks9pPWoTUVfmLpdovn699DpLXBSX#scrollTo=X0qT4WQMK-hS.