{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T22:41:29Z","timestamp":1777675289537,"version":"3.51.4"},"reference-count":27,"publisher":"Wiley","issue":"A","license":[{"start":{"date-parts":[[2016,8,26]],"date-time":"2016-08-26T00:00:00Z","timestamp":1472169600000},"content-version":"unspecified","delay-in-days":238,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2016]]},"abstract":"Given a sextic CM field $K$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we give an explicit method for finding all genus-$3$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> hyperelliptic curves defined over $\\mathbb{C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng\u00a0[J.\u00a0Ramanujan Math. Soc.<\/jats:italic> 16 (2001) no.\u00a04, 339\u2013372], we give an algorithm which works in complete generality, for any CM sextic field $K$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field\u00a0$\\mathbb{F}_{p}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo\u00a0$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1112\/s1461157016000322","type":"journal-article","created":{"date-parts":[[2016,8,26]],"date-time":"2016-08-26T15:29:38Z","timestamp":1472225378000},"page":"283-300","source":"Crossref","is-referenced-by-count":17,"title":["Constructing genus-3 hyperelliptic Jacobians with CM"],"prefix":"10.1112","volume":"19","author":[{"given":"Jennifer S.","family":"Balakrishnan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sorina","family":"Ionica","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christelle","family":"Vincent","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2016,8,26]]},"reference":[{"key":"S1461157016000322_r21","unstructured":"21. 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Cosset , \u2018Applications des fonctions th\u00eata \u00e0 la cryptographie sur les courbes hyperelliptiques\u2019, PhD Thesis, Universit\u00e9 Henri Poincar\u00e9 \u2013 Nancy I, 2011."},{"key":"S1461157016000322_r9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01342938"},{"key":"S1461157016000322_r25","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01020-0"},{"key":"S1461157016000322_r6","volume-title":"Algorithmic number theory: 7th international symposium, ANTS VII","author":"Diem","year":"2006"},{"key":"S1461157016000322_r16","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4578-6"},{"key":"S1461157016000322_r22","unstructured":"22. W. A. Stein , \u2018Sage Mathematics Software (Version 6.10)\u2019, The Sage Development Team, 2015, http:\/\/www.sagemath.org."},{"key":"S1461157016000322_r23","unstructured":"23. M. Streng , \u2018Complex multiplication of abelian surfaces\u2019, PhD Thesis, Universiteit Leiden, 2010."},{"key":"S1461157016000322_r12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65315-5"},{"key":"S1461157016000322_r8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-06-01900-4"},{"key":"S1461157016000322_r11","doi-asserted-by":"publisher","DOI":"10.2307\/2373243"},{"key":"S1461157016000322_r5","doi-asserted-by":"publisher","DOI":"10.1112\/S1461157014000370"},{"key":"S1461157016000322_r19","volume-title":"Complex multiplication of abelian varieties and its applications to number theory","author":"Shimura","year":"1961"},{"key":"S1461157016000322_r2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-06307-1"},{"key":"S1461157016000322_r1","unstructured":"1. J.\u00a0S. Balakrishnan , S. Ionica , K. Lauter and C. 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