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A commensurable class of closed manifolds is achiral if it contains an achiral element, or equivalently, each manifold in $\\CM$ has an achiral finite cover.\n  Each commensurable class containing non-orientable elements must be achiral.\n  It is natural to wonder how many\n  commensurable classes are achiral and how many achiral classes have non-orientable elements.\n  We study this problem for Sol 3-manifolds. Each commensurable class $\\CM$ of Sol 3-manifold has a complete topological invariant $D_{"},"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":"2406.13241","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.GT","submitted_at":"2024-06-19T06:06:38Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cb5ae9c898646ab5202edfa1ff01d9b73b85513bc94be1264125803f8d9c7613","abstract_canon_sha256":"d7d57638871d095da9c52d53725832d3b7a0994c1ced3e4e72dfe65d60f728c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:34:27.294981Z","signature_b64":"kLeDhsuJMmS1nUib/RwkU+7kQ+S2ppGwTyBiKaWkuIsLl2uG0+wXOwq0kz1rn6m4MLYLMMmaWh52TN7bDmbEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72e8df65e70b316416635ac9f22cc740938127b62b6f722d6ca287101505b92f","last_reissued_at":"2026-07-05T08:34:27.294566Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:34:27.294566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Achirality of Sol 3-Manifolds, Stevenhagen Conjecture and Shimizu's L-series","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GT","authors_text":"Shicheng Wang, Ye Tian, Zhongzi Wang","submitted_at":"2024-06-19T06:06:38Z","abstract_excerpt":"A closed orientable manifold is {\\em achiral} if it admits an orientation reversing homeomorphism. 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