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We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\\mathcal{M},\\mathfrak{n})$ where $\\mathcal{M}$'s are isotopes of $\\mathcal{J}$ and $\\mathfrak{n}$'s are cubic \\'etale subalgebras of $\\mathcal{M}$. Also we prove "},"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":"1603.00739","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-02T15:12:10Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"1511305bbe20e5649ca86b47c84c4ac57f2fbe571433883c3d71bc759d6ec254","abstract_canon_sha256":"a85b7ce6d4da1912aaba2f0db082ed410d01c2be85c5618716d0b59cd76cf14a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:43.075051Z","signature_b64":"pWyduL8DRn0+BlFZOMRJs+PJO44lUR9ufOt3RSe6xGafSpaCqVAVMUw288/VNytOCka2+xozstCfMX+pgrouDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4ffff4a5cbdb31919262f44e7a18c772a5aa06b7add9918cd5081ccddf1195c","last_reissued_at":"2026-05-18T01:19:43.074419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:43.074419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational orbits of the space of pairs of exceptional Jordan algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.NT","authors_text":"Akihiko Yukie, Ryo Kato","submitted_at":"2016-03-02T15:12:10Z","abstract_excerpt":"Let $k$ be a field of characteristic not equal to $2,3$, $\\mathbb{O}$ an octonion over $k$ and $\\mathcal{J}$ the exceptional Jordan algebra defined by $\\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\\times \\mathrm{GL}(2)$ and $V=\\mathcal{J}\\oplus \\mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\\mathcal{M},\\mathfrak{n})$ where $\\mathcal{M}$'s are isotopes of $\\mathcal{J}$ and $\\mathfrak{n}$'s are cubic \\'etale subalgebras of $\\mathcal{M}$. Also we prove "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00739","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.00739","created_at":"2026-05-18T01:19:43.074516+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.00739v1","created_at":"2026-05-18T01:19:43.074516+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.00739","created_at":"2026-05-18T01:19:43.074516+00:00"},{"alias_kind":"pith_short_12","alias_value":"WT776SS4XWZR","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"WT776SS4XWZRSGJG","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"WT776SS4","created_at":"2026-05-18T12:30:51.357362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4","json":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4.json","graph_json":"https://pith.science/api/pith-number/WT776SS4XWZRSGJGF5COPIMMO4/graph.json","events_json":"https://pith.science/api/pith-number/WT776SS4XWZRSGJGF5COPIMMO4/events.json","paper":"https://pith.science/paper/WT776SS4"},"agent_actions":{"view_html":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4","download_json":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4.json","view_paper":"https://pith.science/paper/WT776SS4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.00739&json=true","fetch_graph":"https://pith.science/api/pith-number/WT776SS4XWZRSGJGF5COPIMMO4/graph.json","fetch_events":"https://pith.science/api/pith-number/WT776SS4XWZRSGJGF5COPIMMO4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4/action/storage_attestation","attest_author":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4/action/author_attestation","sign_citation":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4/action/citation_signature","submit_replication":"https://pith.science/pith/WT776SS4XWZRSGJGF5COPIMMO4/action/replication_record"}},"created_at":"2026-05-18T01:19:43.074516+00:00","updated_at":"2026-05-18T01:19:43.074516+00:00"}