{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T16:56:33Z","timestamp":1758473793136},"reference-count":13,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,11,7]],"date-time":"2013-11-07T00:00:00Z","timestamp":1383782400000},"content-version":"unspecified","delay-in-days":37,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2013,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We determine the conditions under which singular values of multiple <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S146115701300020X_inline1\" \/><jats:tex-math>$\\eta $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-quotients of square-free level, not necessarily prime to\u00a0six, yield class invariants; that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S146115701300020X_inline2\" \/><jats:tex-math>${2}^{{k}^{\\prime } - 1} $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S146115701300020X_inline3\" \/><jats:tex-math>${k}^{\\prime } \\geq 2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S146115701300020X_inline4\" \/><jats:tex-math>${ X}_{0}^{+ } (p)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S146115701300020X_inline5\" \/><jats:tex-math>$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> prime and ramified.<\/jats:p>","DOI":"10.1112\/s146115701300020x","type":"journal-article","created":{"date-parts":[[2013,11,7]],"date-time":"2013-11-07T14:14:02Z","timestamp":1383833642000},"page":"407-418","source":"Crossref","is-referenced-by-count":4,"title":["Singular values of multiple eta-quotients for ramified primes"],"prefix":"10.1112","volume":"16","author":[{"given":"Andreas","family":"Enge","sequence":"first","affiliation":[]},{"given":"Reinhard","family":"Schertz","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2013,11,7]]},"reference":[{"key":"S146115701300020X_r12","doi-asserted-by":"publisher","DOI":"10.5802\/jtnb.361"},{"key":"S146115701300020X_r13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511776892"},{"key":"S146115701300020X_r2","first-page":"21","volume-title":"Computational perspectives on number theory: proceedings of a conference in honor of A. 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Morain , \u2018Advances in the CM method for elliptic curves\u2019, Slides of Fields Cryptography Retrospective Meeting, May 11\u201315, 2009, http:\/\/www.lix.polytechnique.fr\/~morain\/Exposes\/fields09.pdf."},{"key":"S146115701300020X_r9","doi-asserted-by":"publisher","DOI":"10.1145\/384101.384125"},{"key":"S146115701300020X_r10","doi-asserted-by":"publisher","DOI":"10.5802\/jtnb.143"},{"key":"S146115701300020X_r5","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-44828-4_27"},{"key":"S146115701300020X_r4","unstructured":"4. A. Enge and F. Morain , Generalised Weber functions. Technical Report 385608, HAL-INRIA, 2009, http:\/\/hal.inria.fr\/inria-00385608."}],"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\/S146115701300020X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T02:28:19Z","timestamp":1559874499000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S146115701300020X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10]]},"references-count":13,"alternative-id":["S146115701300020X"],"URL":"https:\/\/doi.org\/10.1112\/s146115701300020x","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10]]}}}