{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:P7OJ6C3DIOA5V4BPAQXGNSDTV4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"caab5401a3958a4216445982f6ff8cd5526006afe04a2c180b39341d4be0fcb0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-21T00:26:54Z","title_canon_sha256":"865c1b3247064878800e81d6bd08a02d6e999b8cedfb327c0208bb2f91b8c349"},"schema_version":"1.0","source":{"id":"1206.4746","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4746","created_at":"2026-05-18T01:22:26Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4746v2","created_at":"2026-05-18T01:22:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4746","created_at":"2026-05-18T01:22:26Z"},{"alias_kind":"pith_short_12","alias_value":"P7OJ6C3DIOA5","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"P7OJ6C3DIOA5V4BP","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"P7OJ6C3D","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:3beca747dc272f3c74744dc45c780ac5518d7d0fc7a5cf86547f1f5b92777980","target":"graph","created_at":"2026-05-18T01:22:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For a binary quadratic form $Q$, we consider the action of $\\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed \"lattice shape\". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with","authors_text":"Ariel Shnidman, Manjul Bhargava","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-21T00:26:54Z","title":"On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4746","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fad262fa8b172710cfcb33583ea42222e26bae9aef4c1eb910c26d06baebbcb9","target":"record","created_at":"2026-05-18T01:22:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"caab5401a3958a4216445982f6ff8cd5526006afe04a2c180b39341d4be0fcb0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-21T00:26:54Z","title_canon_sha256":"865c1b3247064878800e81d6bd08a02d6e999b8cedfb327c0208bb2f91b8c349"},"schema_version":"1.0","source":{"id":"1206.4746","kind":"arxiv","version":2}},"canonical_sha256":"7fdc9f0b634381daf02f042e66c873af2fccc07a0e2cf77347e51df22e295994","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7fdc9f0b634381daf02f042e66c873af2fccc07a0e2cf77347e51df22e295994","first_computed_at":"2026-05-18T01:22:26.564327Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:26.564327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qzkgLCnDZp4LbwizqMVdH6xhAvHBAtzeaXcL3sJDWVOC0/sq2ERXI/j4RF8DQ35tFn4lq9eLmWQxE+h2Y7LbDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:26.565178Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.4746","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fad262fa8b172710cfcb33583ea42222e26bae9aef4c1eb910c26d06baebbcb9","sha256:3beca747dc272f3c74744dc45c780ac5518d7d0fc7a5cf86547f1f5b92777980"],"state_sha256":"37843092a578e45ec54c4cf445b9c2236ee40dd57c6a735d988810d18e1bfa75"}