{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T23:24:01Z","timestamp":1783985041575,"version":"3.55.0"},"reference-count":35,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2014,9,1]],"date-time":"2014-09-01T00:00:00Z","timestamp":1409529600000},"content-version":"unspecified","delay-in-days":243,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We use an invariant-theoretic method to compute certain twists of the modular curves <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline1\"\/><jats:tex-math>$\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}X(n)$<\/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\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline2\"\/><jats:tex-math>$n=7,11$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Searching for rational points on these twists enables us to find non-trivial pairs 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=\"S1461157014000059_inline3\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-congruent elliptic curves over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline4\"\/><jats:tex-math>${\\mathbb{Q}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, that is, pairs of non-isogenous elliptic curves over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline5\"\/><jats:tex-math>${\\mathbb{Q}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> whose <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline6\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline7\"\/><jats:tex-math>${\\mathbb{Q}}(T)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves\u00a0 over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000059_inline8\"\/><jats:tex-math>${\\mathbb{Q}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p><jats:p><jats:uri xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:type=\"simple\" xlink:href=\"http:\/\/journals.cambridge.org\/sup_S1461157014000059sup001\">Supplementary\u00a0materials\u00a0are\u00a0available\u00a0with\u00a0this\u00a0article.<\/jats:uri><\/jats:p>","DOI":"10.1112\/s1461157014000059","type":"journal-article","created":{"date-parts":[[2014,10,7]],"date-time":"2014-10-07T08:28:37Z","timestamp":1412670517000},"page":"536-564","source":"Crossref","is-referenced-by-count":12,"title":["On families of 7- and 11-congruent elliptic curves"],"prefix":"10.1112","volume":"17","author":[{"given":"Tom","family":"Fisher","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2014,9,1]]},"reference":[{"key":"S1461157014000059_r28","first-page":"148","volume-title":"Elliptic curves, modular forms and Fermat\u2019s Last Theorem, Hong Kong, 1993","volume":"I","author":"Rubin","year":"1995"},{"key":"S1461157014000059_r34","article-title":"Courbes elliptique munies d\u2019un sous-group \u2124\u2215n\u2124 \u00d7 \u03bc\n                  \n                     n","volume":"57","author":"V\u00e9lu","year":"1978","journal-title":"M\u00e9m. 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