{"id":184601,"date":"2024-03-09T02:00:37","date_gmt":"2024-03-09T08:00:37","guid":{"rendered":"https:\/\/lifeboat.com\/blog\/2024\/03\/elliptic-curve-murmurations-found-with-ai-take-flight"},"modified":"2024-03-09T02:00:37","modified_gmt":"2024-03-09T08:00:37","slug":"elliptic-curve-murmurations-found-with-ai-take-flight","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2024\/03\/elliptic-curve-murmurations-found-with-ai-take-flight","title":{"rendered":"Elliptic Curve \u2018Murmurations\u2019 Found With AI Take Flight"},"content":{"rendered":"<p><\/p>\n<p><iframe style=\"display: block; margin: 0 auto; width: 100%; aspect-ratio: 4\/3; object-fit: contain;\" src=\"https:\/\/www.youtube.com\/embed\/CEl2W2A5Z4o?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope;\n   picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>Draw a line between <em>P<\/em> and <em>Q<\/em>. That line will intersect the curve at a third point, <em>R<\/em>. (Mathematicians have a special trick for dealing with the case where the line doesn\u2019t intersect the curve by adding a \u201cpoint at infinity.\u201d) The reflection of <em>R<\/em> across the <em>x<\/em>-axis is your sum <em>P<\/em> + <em>Q<\/em>. Together with this addition operation, all the solutions to the curve form a mathematical object called a group.<\/p>\n<p>Mathematicians use this to define the \u201crank\u201d of a curve. The <a href=\"https:\/\/www.quantamagazine.org\/without-a-proof-mathematicians-wonder-how-much-evidence-is-enough-20181031\/\">rank of a curve<\/a> relates to the number of rational solutions it has. Rank 0 curves have a finite number of solutions. Curves with higher rank have infinite numbers of solutions whose relationship to one another using the addition operation is described by the rank.<\/p>\n<p>Ranks are not well understood; mathematicians don\u2019t always have a way of computing them and don\u2019t know how big they can get. (The largest exact rank known for a specific curve is 20.) Similar-looking curves can have completely different ranks.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Draw a line between P and Q. That line will intersect the curve at a third point, R. (Mathematicians have a special trick for dealing with the case where the line doesn\u2019t intersect the curve by adding a \u201cpoint at infinity.\u201d) The reflection of R across the x-axis is your sum P + Q. Together [\u2026]<\/p>\n","protected":false},"author":427,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2229,6],"tags":[],"class_list":["post-184601","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-robotics-ai"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/184601","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/427"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=184601"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/184601\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=184601"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=184601"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=184601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}