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Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\\mathscr{O}$ characterize the isomorphism class of $\\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\\mathbb{F}_p$ or in the set $\\mathbb{F}_{p^2} \\setminus"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2503.03478","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2025-03-05T13:15:24Z","cross_cats_sorted":[],"title_canon_sha256":"97e53313f8103b313acf456a1157892e0a3c61640cadb7d7911c0ca2c8cb6cae","abstract_canon_sha256":"59ad045d4086e6c83ab50130fb1fd56f47715be03d8da45c7733235f5af28228"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:04:13.260926Z","signature_b64":"jd7hKnKKeIV77xR8e2wTG2ZzRhHsPKBZT5k89HU/JffPio6NKXOlMW0MYtsiqxpA31T/xHV3kfT03QWe5dhxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cd027ced04268faf6847a6fb23727d753c71fd3fa4242e7e20dc6ff01ebef75","last_reissued_at":"2026-05-21T01:04:13.260077Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:04:13.260077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gross lattices of supersingular elliptic curves","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chenfeng He, Christelle Vincent, Gaurish Korpal, Ha T. 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