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This density is determined by the image of the arboreal Galois representation $\\tau_{E,2^{k}} : {\\rm Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q}) \\to {\\rm AGL}_{2}(\\mathbb{Z}/2^{k}\\mathbb{Z})$. Assuming that $\\alpha$ is primitive (that is, neither $\\alpha$ nor $\\alpha + T$ is twice a point over $\\mathbb{Q}$) and that the image of the "},"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":"1810.10583","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-24T19:22:35Z","cross_cats_sorted":[],"title_canon_sha256":"fa581955dd663d5d890816462b025f071f45bdfc4cd1b035ccdb02f4a1eceaa4","abstract_canon_sha256":"0060a95c43ff075b783ea5166119f6206cc26315cbde3d0632f3d99de044ab33"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:11.372003Z","signature_b64":"X79c5rOpoI2y1SJVLsjEuxrj+wQrwbplvMwzrEZfmSgUrzPKJ1/RFkev89u8Rwys+P7ZY8J9yMA3K/tM1c92Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6036aad34cc5db08986ee98cf99cb697cb9dbd1be9a1ffa201eeb2c45fa064a1","last_reissued_at":"2026-05-17T23:49:11.371423Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:11.371423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The density of odd order reductions for elliptic curves with a rational point of order 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeremy Rouse, Ke Liang","submitted_at":"2018-10-24T19:22:35Z","abstract_excerpt":"Suppose that $E/\\mathbb{Q}$ is an elliptic curve with a rational point $T$ of order $2$ and $\\alpha \\in E(\\mathbb{Q})$ is a point of infinite order. We consider the problem of determining the density of primes $p$ for which $\\alpha \\in E(\\mathbb{F}_{p})$ has odd order. This density is determined by the image of the arboreal Galois representation $\\tau_{E,2^{k}} : {\\rm Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q}) \\to {\\rm AGL}_{2}(\\mathbb{Z}/2^{k}\\mathbb{Z})$. 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