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The best known upper bound is $N(5,D_5,X)\\ll X^{3/4 + \\epsilon}$, and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is $\\ll X^{2/3}$. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for "},"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":"1107.4111","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-20T20:21:55Z","cross_cats_sorted":[],"title_canon_sha256":"86781d89afc264a2942cf3523cce5ffe8ba0d2c15187634cea5214631884e4c5","abstract_canon_sha256":"76f8b95447107f2da53e70778d21dbfdcad359fa299a96efb4053a7b2f97e3d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:26.975757Z","signature_b64":"hTMie/EeGx5wzb5ldxlygL/uVnCx1iK6QVcXKe2ol5xrTIDHEJncHaRUcqSNNJtqfoWdO9Ugn3ixZapevXqjDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d52c81031377231222f59a8c2f7d758d7cf68ad4af2472350e2a6b5b237bb12","last_reissued_at":"2026-05-18T04:09:26.975217Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:26.975217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Progress Towards Counting D_5 Quintic Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Larson, Larry Rolen","submitted_at":"2011-07-20T20:21:55Z","abstract_excerpt":"Let $N(5,D_5,X)$ be the number of quintic number fields whose Galois closure has Galois group $D_5$ and whose discriminant is bounded by $X$. 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