The Level 12 Eisenstein Cyclotomic Node: A Direct Genus-Two Quotient and an Independent Four-Cover Chabauty Verification

This preprint determines the complete rational point set of the affinegenus-three curve arising from the Eisenstein norm-square condition on thePell-Pythagorean orbit generated by 2+sqrt(3). An explicit involution givesa genus-two quotient whose rational points are determined completely by acertified Magma computation. Exact lifting yields eight rational points onthe original curve and proves that the unique positive Pell index is n=2,corresponding to the primitive Pythagorean triple (33,56,65) and H_2=61^2.An independent four-cover descent and classical Chabauty verification isincluded, together with executable Magma scripts and matching output logs.The genus-two quotient used in the principal proof was identified by JohnVoight and is acknowledged in the manuscript.