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Following Weierstrass for the hyperelliptic case, we define an ``$\\mathrm{al}$'' function for this curve and $\\mathrm{al}^{(c)}_r$, $c=0,1,2$, for each one of three particular covers of the Jacobian of the curve, and $r=1,2,3,4$ for a finite branchpoint $(b_r,0)$. This generalization of the Jacobi $\\mathrm{sn}$, $\\mathrm{cn}$, $\\mathrm{dn}$ functions satisfies the relation: $$ \\sum_{r=1}^"},"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":"1312.4107","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-12-15T04:55:36Z","cross_cats_sorted":["nlin.SI"],"title_canon_sha256":"cc29cbeb1b7635cb69670f4fe08cd1b7ae60aedba33c073851def30506602a18","abstract_canon_sha256":"0d56fd30c5b17ad575a2f02ae09382543502cc476e2f2678daf0d5cdb74689d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:35.688969Z","signature_b64":"v6YAiLxN9aTmWZGUppvBhLlAJt5JDWza/4Fxt7fa7GLbc+wFBLPEcQnDe6U5dNVlbb4F2+Phex327/GmuRLiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1813fe4f50bef877bd1cbeb9b74761c1c0d4c36b459662a1ec609a601989faaa","last_reissued_at":"2026-05-18T03:04:35.688193Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:35.688193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The al function of a cyclic trigonal curve of genus three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.SI"],"primary_cat":"math.AG","authors_text":"Emma Previato, Shigeki Matsutani","submitted_at":"2013-12-15T04:55:36Z","abstract_excerpt":"A cyclic trigonal curve of genus three is a $\\mathbb{Z}_3$ Galois cover of $\\mathbb{P}^1$, therefore can be written as a smooth plane curve with equation $y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. 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