{"title":"高阶椭圆曲线与扭转ℤ/4ℤ。","authors":"Foad Khoshnam, Dustin Moody","doi":"","DOIUrl":null,"url":null,"abstract":"<p><p>Working over the field ℚ(<i>t</i>), Kihara constructed an elliptic curve with torsion group ℤ/4ℤ and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group ℤ/4ℤ and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high ranks 10 and 11.</p>","PeriodicalId":36228,"journal":{"name":"Integers","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5535278/pdf/nihms875075.pdf","citationCount":"0","resultStr":"{\"title\":\"High rank elliptic curves with torsion ℤ/4ℤ.\",\"authors\":\"Foad Khoshnam, Dustin Moody\",\"doi\":\"\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Working over the field ℚ(<i>t</i>), Kihara constructed an elliptic curve with torsion group ℤ/4ℤ and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group ℤ/4ℤ and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high ranks 10 and 11.</p>\",\"PeriodicalId\":36228,\"journal\":{\"name\":\"Integers\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5535278/pdf/nihms875075.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Integers\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Integers","FirstCategoryId":"1085","ListUrlMain":"","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
Working over the field ℚ(t), Kihara constructed an elliptic curve with torsion group ℤ/4ℤ and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group ℤ/4ℤ and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high ranks 10 and 11.