{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:58:39Z","timestamp":1776848319806,"version":"3.51.2"},"reference-count":27,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2015,8,1]],"date-time":"2015-08-01T00:00:00Z","timestamp":1438387200000},"content-version":"unspecified","delay-in-days":212,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>We study elliptic curves over quadratic fields with isogenies of certain degrees. Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline2\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be a positive integer such that the modular curve <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline3\"\/><jats:tex-math>$X_{0}(n)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is hyperelliptic of genus <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline4\"\/><jats:tex-math>${\\geqslant}2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and such that its Jacobian has rank\u00a0<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline5\"\/><jats:tex-math>$0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> over\u00a0<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline6\"\/><jats:tex-math>$\\mathbb{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We determine all points of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline7\"\/><jats:tex-math>$X_{0}(n)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline8\"\/><jats:tex-math>$\\overline{\\mathbb{Q}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-isomorphism, every elliptic curve over a quadratic field\u00a0<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline9\"\/><jats:tex-math>$K$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> admitting an <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline10\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-isogeny is <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline11\"\/><jats:tex-math>$d$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-isogenous, for some <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline12\"\/><jats:tex-math>$d\\mid n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, to the twist of its Galois conjugate by a quadratic extension <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline13\"\/><jats:tex-math>$L$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of\u00a0<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline14\"\/><jats:tex-math>$K$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We determine <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline15\"\/><jats:tex-math>$d$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and\u00a0<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline16\"\/><jats:tex-math>$L$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline17\"\/><jats:tex-math>$\\overline{\\mathbb{Q}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-isomorphism, all elliptic curves with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline18\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-isogenies over quadratic fields are in fact <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000157_inline19\"\/><jats:tex-math>$\\mathbb{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-curves.<\/jats:p>","DOI":"10.1112\/s1461157015000157","type":"journal-article","created":{"date-parts":[[2015,8,25]],"date-time":"2015-08-25T00:29:16Z","timestamp":1440462556000},"page":"578-602","source":"Crossref","is-referenced-by-count":28,"title":["Hyperelliptic modular curves  and isogenies of elliptic curves over quadratic fields"],"prefix":"10.1112","volume":"18","author":[{"given":"Peter","family":"Bruin","sequence":"first","affiliation":[]},{"given":"Filip","family":"Najman","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2015,8,1]]},"reference":[{"key":"S1461157015000157_r1","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1998.2343"},{"key":"S1461157015000157_r17","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-23.3.415"},{"key":"S1461157015000157_r14","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100055444"},{"key":"S1461157015000157_r26","doi-asserted-by":"publisher","DOI":"10.4064\/aa98-3-4"},{"key":"S1461157015000157_r8","doi-asserted-by":"publisher","DOI":"10.5802\/aif.1273"},{"key":"S1461157015000157_r6","first-page":"21","volume-title":"Computational perspectives on number theory (Chicago, IL, 1995)","author":"Elkies","year":"1998"},{"key":"S1461157015000157_r19","doi-asserted-by":"publisher","DOI":"10.1017\/S0027763000002816"},{"key":"S1461157015000157_r11","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610706022940"},{"key":"S1461157015000157_r9","doi-asserted-by":"publisher","DOI":"10.1007\/PL00000508"},{"key":"S1461157015000157_r21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01390348"},{"key":"S1461157015000157_r22","doi-asserted-by":"publisher","DOI":"10.1007\/s002220050059"},{"key":"S1461157015000157_r15","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100056462"},{"key":"S1461157015000157_r18","doi-asserted-by":"publisher","DOI":"10.1016\/0022-314X(82)90025-7"},{"key":"S1461157015000157_r4","doi-asserted-by":"publisher","DOI":"10.1093\/imrn\/rnt013"},{"key":"S1461157015000157_r7","unstructured":"7. 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