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We prove that if $K/\\mathbb{Q}_p$ is unramified and $E$ has additive, potentially supersingular reduction, then $\\alpha_{\\phi/K}$ is determined by the number of distinct geometric components on the special fibers of the minimal proper regular models of $E$ and $E^\\prime$."},"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":"1703.02148","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-06T23:47:10Z","cross_cats_sorted":[],"title_canon_sha256":"89407f312f6f48d1c21d21268151ec02f68991154159b30e6ca203ba6c6a24f1","abstract_canon_sha256":"1dee66e8326a2eaa7458e1b2eaa28df95ba79b15e92b8d21532f3429e7707dae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:20.635031Z","signature_b64":"dKRGeLpJbO+xgxMDj73C6eMfodABOKhWy+AypoI7ouovmpNMklXyelmly51SKkPfYh52Uggq0Mm1Xb+6oqXoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2c848ef72680ea39520dd03446a2bedbdc79506945cb8b301a06d568e5e4e22","last_reissued_at":"2026-05-18T00:49:20.634627Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:20.634627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a local invariant of elliptic curves with a p-isogeny","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matthew Gealy, Zev Klagsbrun","submitted_at":"2017-03-06T23:47:10Z","abstract_excerpt":"An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $\\phi:E\\rightarrow E^\\prime$ comes equipped with an invariant $\\alpha_{\\phi/K}$ that measures the valuation of the leading term of the formal group homomorphism $\\Phi:\\hat E \\rightarrow \\hat E^\\prime$. 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