{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:43:42Z","timestamp":1775839422158,"version":"3.50.1"},"reference-count":18,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":396,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2009]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>If (<jats:italic>Q<\/jats:italic>,<jats:italic>A<\/jats:italic>) is a marked polygon with one interior point, then a general polynomial<jats:italic>f<\/jats:italic>belonging to<jats:bold>K<\/jats:bold>[<jats:italic>x,y<\/jats:italic>] with support<jats:italic>A<\/jats:italic>defines an elliptic curve<jats:italic>C<jats:sub>f<\/jats:sub><\/jats:italic>on the toric surface<jats:italic>X<jats:sub>A<\/jats:sub><\/jats:italic>. If<jats:bold>K<\/jats:bold>has a non-archimedean valuation into<jats:bold>R<\/jats:bold>we can tropicalize<jats:italic>C<jats:sub>f<\/jats:sub><\/jats:italic>to get a tropical curve Trop(<jats:italic>C<jats:sub>f<\/jats:sub><\/jats:italic>). If in the Newton subdivision induced by<jats:italic>f<\/jats:italic>is a triangulation and the interior point occurs as the vertex of a triangle, then Trop(<jats:italic>C<jats:sub>f<\/jats:sub><\/jats:italic>) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of the<jats:italic>j<\/jats:italic>-invariant of<jats:italic>C<jats:sub>f<\/jats:sub><\/jats:italic>.<\/jats:p>","DOI":"10.1112\/s1461157000001522","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T11:42:44Z","timestamp":1375789364000},"page":"275-294","source":"Crossref","is-referenced-by-count":8,"title":["The Tropical<i>j<\/i>-Invariant"],"prefix":"10.1112","volume":"12","author":[{"given":"Eric","family":"Katz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hannah","family":"Markwig","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"Markwig","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000001522_ref018","unstructured":"18. Vigeland Magnus Dehli , \u2018The group law on a tropical elliptic curve.\u2019 Math.AG\/0411485."},{"key":"S1461157000001522_ref014","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.2000.12005186"},{"key":"S1461157000001522_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.08.018"},{"key":"S1461157000001522_ref008","unstructured":"8. Jensen Anders Nedergaard , Markwig Hannah and Markwig Thomas , \u2018tropical.lib.\u2019A singular 3.0 library for computations in tropical geometry, (2007). http:\/\/www.singular.uni-kl.de\/~keilen\/de\/tropical.html."},{"key":"S1461157000001522_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"S1461157000001522_ref006","unstructured":"6. Greuel G.-M. , Pfister G. and Sch\u00f6nemann H. , \u2018Singular 3.0.\u2019 A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, (2005). http:\/\/www.singular.uni-kl.de."},{"key":"S1461157000001522_ref004","unstructured":"4. Gawrilow Ewgenij and Joswig Michael , \u2018polymake 2.3.\u2019Tech. rep., TU Berlin and TU Darmstadt, (1997). http:\/\/www.math.tu-berlin.de\/polymake."},{"key":"S1461157000001522_ref003","doi-asserted-by":"crossref","DOI":"10.1515\/9781400882526","volume-title":"Introduction to toric varieties","author":"Fulton","year":"1993"},{"key":"S1461157000001522_ref015","first-page":"83","article-title":"A census of convex lattice polygons with at most one interior lattice point","volume":"28","author":"Rabinowitz","year":"1989","journal-title":"Ars Combin"},{"key":"S1461157000001522_ref010","unstructured":"10. Katz Eric , Markwig Hannah and Markwig Thomas , \u2018jinvariant.lib.\u2019 A singular 3.0 library for computations with j-invariants in tropical geometry, (2007). http:\/\/www.singular.uni-kl.de\/~keilen\/de\/jinvariant.html."},{"key":"S1461157000001522_ref016","first-page":"330","volume-title":"Mathematical Software - ICMS 2002","author":"Rambau","year":"2002"},{"key":"S1461157000001522_ref001","unstructured":"1. Allermann Lars and Rau Johannes , \u2018First steps in tropical intersection theory.\u2019 ArXiv:0709.3705."},{"key":"S1461157000001522_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4771-1"},{"key":"S1461157000001522_ref017","unstructured":"17. Speyer David ,\u2018Uniformizing tropical curves I:Genus zero and one.\u2019 ArXiv:0711.2677."},{"key":"S1461157000001522_ref011","unstructured":"11. Kerber Michael and Markwig Hannah , \u2018Counting tropical elliptic curves with fixed j-invariant.\u2019 To appear in Commentarii Mathematici Helvetici. Math.AG\/0608472."},{"key":"S1461157000001522_ref013","unstructured":"13. Mikhalkin Grigory , \u2018Tropical geometry and its applications.\u2019 International Congress of Mathematicians, Eur. Math. Soc., vol. II (2006) pp. 827\u2013852."},{"key":"S1461157000001522_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-05-00477-7"},{"key":"S1461157000001522_ref002","first-page":"139","article-title":"Non-archimedean amoebas and tropical varieties","volume":"601","author":"Einsiedler","year":"2006","journal-title":"J. Reine Angew. Math."}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157000001522","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T13:53:55Z","timestamp":1596290035000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157000001522\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009]]},"references-count":18,"alternative-id":["S1461157000001522"],"URL":"https:\/\/doi.org\/10.1112\/s1461157000001522","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009]]}}}