1234:
909:
578:
2148:
Bos, Joppe W.; Halderman, J. Alex; Heninger, Nadia; Moore, Jonathan; Naehrig, Michael; Wustrow, Eric (2014). "Elliptic Curve
Cryptography in Practice". In Christin, Nicolas; Safavi-Naini, Reihaneh (eds.).
453:
251:
2263:
1706:
1869:
with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.
1756:
684:
2128:
1369:
1841:, one is its quadratic twist, and only the other two are really new. Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
2087:
2013:
977:
2038:
1964:
1597:
280:
938:
1783:
1572:
1540:
1493:
1284:
1052:
628:
351:
1060:
710:
of the curves is the same; and so a family of curves related by twisting becomes a useful setting in which to study the arithmetic properties of elliptic curves.
2107:
2058:
1984:
1939:
1915:
1895:
1867:
1839:
1815:
1617:
1513:
1461:
1441:
1421:
1397:
1304:
1257:
1025:
1001:
775:
751:
731:
704:
648:
601:
324:
300:
152:
128:
104:
2168:
2546:
783:
464:
2443:
359:
160:
1797:
It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve
686:. Qualitatively speaking, the arithmetic of a curve and its quadratic twist can look very different in the field
75:
2541:
107:
1625:
2118:
1786:
980:
2123:
2435:
1711:
2345:
2220:
2155:. Lecture Notes in Computer Science. Vol. 8437. Berlin, Heidelberg: Springer. pp. 157–175.
67:
656:
1312:
71:
32:
2520:
2466:
2361:
2335:
2323:
2301:
2275:
2244:
2210:
1918:
24:
2063:
1989:
943:
2409:
2293:
2236:
2164:
44:
2387:
1229:{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+(a_{2}+da_{1}^{2})x^{2}+a_{4}x+a_{6}+da_{3}^{2}.\,}
259:
2510:
2458:
2399:
2383:
2353:
2285:
2228:
2156:
917:
707:
2484:
2421:
1761:
1545:
1518:
1466:
1262:
1030:
606:
329:
2417:
1307:
651:
51:
of degree 1, that is an invertible isogeny. Some curves have higher order twists such as
2349:
2224:
2150:
2018:
1944:
1577:
2092:
2043:
1969:
1924:
1900:
1880:
1852:
1824:
1800:
1602:
1498:
1446:
1426:
1406:
1382:
1289:
1242:
1010:
986:
760:
754:
736:
716:
689:
633:
586:
309:
285:
137:
131:
113:
89:
28:
2515:
2498:
2462:
2535:
2365:
2198:
2357:
2305:
2248:
1400:
2404:
2289:
2160:
2439:
2190:
60:
40:
20:
2379:
2232:
2194:
2499:"On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms"
2413:
2297:
2240:
2524:
2470:
48:
1877:
Twists can be defined for other smooth projective curves as well. Let
2280:
904:{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}
2340:
2215:
1599:
on that same curve (which can happen if the characteristic is not
47:
of K. In particular, an isomorphism between elliptic curves is an
2262:
Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2007-03-15).
573:{\displaystyle y^{2}=x^{3}+da_{2}x^{2}+d^{2}a_{4}x+d^{3}a_{6}.\,}
2199:"Ranks of twists of elliptic curves and Hilbert's tenth problem"
16:
Mathematical curves that are isomorphic over algebraic closures
1941:
that is irreducible and geometrically connected. Then a twist
1849:
Analogously to the quartic twist case, an elliptic curve over
66:
Applications of twists include cryptography, the solution of
1986:
is another smooth projective curve for which there exists a
1821:, one obtains precisely four curves: one is isomorphic to
2322:
Lombardo, Davide; Lorenzo GarcĂa, Elisa (February 2019).
1574:
is on just one of the curves, there is exactly one other
2444:"The square-free sieve and the rank of elliptic curves"
448:{\displaystyle dy^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}
246:{\displaystyle y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}
2095:
2066:
2046:
2021:
1992:
1972:
1947:
1927:
1903:
1883:
1855:
1827:
1803:
1764:
1714:
1628:
1605:
1580:
1548:
1521:
1501:
1469:
1449:
1429:
1409:
1385:
1315:
1292:
1265:
1245:
1063:
1033:
1013:
989:
946:
920:
786:
763:
739:
719:
692:
659:
636:
609:
589:
467:
362:
332:
312:
288:
262:
163:
140:
116:
92:
2101:
2081:
2052:
2032:
2007:
1978:
1958:
1933:
1909:
1889:
1861:
1833:
1809:
1777:
1750:
1700:
1611:
1591:
1566:
1534:
1507:
1487:
1455:
1435:
1415:
1391:
1363:
1298:
1278:
1251:
1228:
1046:
1019:
995:
971:
932:
903:
769:
745:
725:
698:
678:
642:
622:
595:
572:
447:
345:
318:
294:
274:
245:
146:
122:
98:
2264:"Twists of X(7) and primitive solutions to x+y=z"
2129:Twisted tripling-oriented Doche–Icart–Kohel curve
713:Twists can also be defined when the base field
2392:Bulletin of the American Mathematical Society
8:
2503:Journal of the American Mathematical Society
2451:Journal of the American Mathematical Society
2324:"Computing twists of hyperelliptic curves"
39:, that is another elliptic curve which is
2514:
2497:C. L. Stewart and J. Top (October 1995).
2403:
2339:
2317:
2315:
2279:
2214:
2094:
2068:
2067:
2065:
2045:
2020:
1994:
1993:
1991:
1971:
1946:
1926:
1902:
1882:
1854:
1826:
1802:
1769:
1763:
1742:
1724:
1719:
1713:
1678:
1663:
1654:
1646:
1629:
1627:
1604:
1579:
1547:
1526:
1520:
1500:
1468:
1448:
1428:
1408:
1384:
1340:
1328:
1314:
1291:
1270:
1264:
1244:
1225:
1216:
1211:
1195:
1179:
1166:
1153:
1148:
1132:
1116:
1100:
1081:
1068:
1062:
1038:
1032:
1012:
988:
951:
945:
919:
900:
891:
875:
862:
852:
839:
823:
804:
791:
785:
762:
738:
718:
691:
666:
658:
635:
614:
608:
588:
569:
560:
550:
534:
524:
511:
501:
485:
472:
466:
444:
435:
419:
406:
396:
383:
370:
361:
337:
331:
311:
287:
261:
242:
233:
217:
204:
194:
181:
168:
162:
139:
115:
91:
59:. The curve and its twists have the same
2152:Financial Cryptography and Data Security
2140:
1701:{\displaystyle |E(K)|+|E^{d}(K)|=2q+2}
7:
1917:be curve over that field, i.e., a
1375:Quadratic twist over finite fields
14:
2516:10.1090/S0894-0347-1995-1290234-5
2463:10.1090/S0894-0347-1991-1080648-7
1751:{\displaystyle t_{E^{d}}=-t_{E}}
2358:10.1016/j.jalgebra.2018.08.035
2073:
1999:
1679:
1675:
1669:
1655:
1647:
1643:
1637:
1630:
1561:
1549:
1482:
1470:
1358:
1333:
1325:
1319:
1159:
1125:
679:{\displaystyle K({\sqrt {d}})}
673:
663:
1:
2405:10.1090/S0273-0979-02-00952-7
2290:10.1215/S0012-7094-07-13714-1
1364:{\displaystyle K/(X^{2}+X+d)}
2161:10.1007/978-3-662-45472-5_11
2089:is the algebraic closure of
733:is of characteristic 2. Let
2547:Elliptic curve cryptography
1054:, defined by the equation:
353:, defined by the equation:
2563:
2388:"Ranks of elliptic curves"
2082:{\displaystyle {\bar {K}}}
2008:{\displaystyle {\bar {K}}}
70:, and when generalized to
2268:Duke Mathematical Journal
2233:10.1007/s00222-010-0252-0
972:{\displaystyle X^{2}+X+d}
2203:Inventiones Mathematicae
1286:are not isomorphic over
1239:The two elliptic curves
630:are not isomorphic over
583:The two elliptic curves
2483:P. Stevenhagen (2008).
1423:elements, then for all
275:{\displaystyle d\neq 0}
2492:. Universiteit Leiden.
2119:Twisted Hessian curves
2103:
2083:
2054:
2034:
2009:
1980:
1960:
1935:
1911:
1891:
1863:
1835:
1811:
1787:Frobenius endomorphism
1779:
1752:
1702:
1613:
1593:
1568:
1536:
1509:
1489:
1457:
1437:
1417:
1393:
1365:
1300:
1280:
1253:
1230:
1048:
1021:
997:
981:irreducible polynomial
973:
934:
933:{\displaystyle d\in K}
905:
771:
747:
727:
700:
680:
650:, but rather over the
644:
624:
597:
574:
449:
347:
320:
296:
276:
247:
148:
124:
110:different from 2. Let
100:
2124:Twisted Edwards curve
2104:
2084:
2055:
2035:
2015:-isomorphism between
2010:
1981:
1961:
1936:
1912:
1892:
1864:
1836:
1812:
1780:
1778:{\displaystyle t_{E}}
1753:
1703:
1614:
1594:
1569:
1567:{\displaystyle (x,y)}
1537:
1535:{\displaystyle E^{d}}
1510:
1490:
1488:{\displaystyle (x,y)}
1458:
1438:
1418:
1394:
1366:
1301:
1281:
1279:{\displaystyle E^{d}}
1254:
1231:
1049:
1047:{\displaystyle E^{d}}
1022:
998:
974:
935:
906:
772:
748:
728:
701:
681:
645:
625:
623:{\displaystyle E^{d}}
598:
575:
450:
348:
346:{\displaystyle E^{d}}
321:
297:
277:
248:
149:
125:
101:
68:Diophantine equations
35:K has an associated
2093:
2064:
2044:
2019:
1990:
1970:
1945:
1925:
1921:of dimension 1 over
1901:
1881:
1853:
1825:
1801:
1785:is the trace of the
1762:
1712:
1626:
1603:
1578:
1546:
1519:
1499:
1467:
1463:such that the point
1447:
1427:
1407:
1383:
1313:
1290:
1263:
1243:
1061:
1031:
1011:
987:
944:
918:
784:
761:
737:
717:
690:
657:
634:
607:
587:
465:
360:
330:
310:
286:
260:
161:
138:
114:
90:
76:Sato–Tate conjecture
72:hyperelliptic curves
2350:2016arXiv161104856L
2225:2010InMat.181..541M
1221:
1158:
74:, the study of the
2328:Journal of Algebra
2197:(September 2010).
2099:
2079:
2060:, where the field
2050:
2033:{\displaystyle C'}
2030:
2005:
1976:
1959:{\displaystyle C'}
1956:
1931:
1919:projective variety
1907:
1887:
1859:
1831:
1807:
1775:
1748:
1698:
1622:As a consequence,
1609:
1592:{\displaystyle y'}
1589:
1564:
1532:
1505:
1495:belongs to either
1485:
1453:
1433:
1413:
1389:
1361:
1296:
1276:
1249:
1226:
1207:
1144:
1044:
1017:
993:
969:
930:
901:
767:
743:
723:
696:
676:
640:
620:
593:
570:
445:
343:
316:
292:
272:
243:
144:
120:
96:
25:algebraic geometry
2384:Silverberg, Alice
2170:978-3-662-45471-8
2102:{\displaystyle K}
2076:
2053:{\displaystyle C}
2002:
1979:{\displaystyle C}
1934:{\displaystyle K}
1910:{\displaystyle C}
1890:{\displaystyle K}
1862:{\displaystyle K}
1834:{\displaystyle E}
1810:{\displaystyle E}
1612:{\displaystyle 2}
1508:{\displaystyle E}
1456:{\displaystyle y}
1436:{\displaystyle x}
1416:{\displaystyle q}
1392:{\displaystyle K}
1299:{\displaystyle K}
1252:{\displaystyle E}
1020:{\displaystyle E}
996:{\displaystyle K}
770:{\displaystyle K}
746:{\displaystyle E}
726:{\displaystyle K}
699:{\displaystyle K}
671:
643:{\displaystyle K}
596:{\displaystyle E}
319:{\displaystyle E}
295:{\displaystyle K}
147:{\displaystyle K}
123:{\displaystyle E}
99:{\displaystyle K}
45:algebraic closure
2554:
2528:
2518:
2493:
2491:
2475:
2474:
2448:
2432:
2426:
2425:
2407:
2376:
2370:
2369:
2343:
2319:
2310:
2309:
2283:
2259:
2253:
2252:
2218:
2187:
2181:
2180:
2178:
2177:
2145:
2108:
2106:
2105:
2100:
2088:
2086:
2085:
2080:
2078:
2077:
2069:
2059:
2057:
2056:
2051:
2039:
2037:
2036:
2031:
2029:
2014:
2012:
2011:
2006:
2004:
2003:
1995:
1985:
1983:
1982:
1977:
1965:
1963:
1962:
1957:
1955:
1940:
1938:
1937:
1932:
1916:
1914:
1913:
1908:
1896:
1894:
1893:
1888:
1868:
1866:
1865:
1860:
1840:
1838:
1837:
1832:
1816:
1814:
1813:
1808:
1784:
1782:
1781:
1776:
1774:
1773:
1757:
1755:
1754:
1749:
1747:
1746:
1731:
1730:
1729:
1728:
1708:or equivalently
1707:
1705:
1704:
1699:
1682:
1668:
1667:
1658:
1650:
1633:
1618:
1616:
1615:
1610:
1598:
1596:
1595:
1590:
1588:
1573:
1571:
1570:
1565:
1541:
1539:
1538:
1533:
1531:
1530:
1514:
1512:
1511:
1506:
1494:
1492:
1491:
1486:
1462:
1460:
1459:
1454:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1398:
1396:
1395:
1390:
1370:
1368:
1367:
1362:
1345:
1344:
1332:
1305:
1303:
1302:
1297:
1285:
1283:
1282:
1277:
1275:
1274:
1258:
1256:
1255:
1250:
1235:
1233:
1232:
1227:
1220:
1215:
1200:
1199:
1184:
1183:
1171:
1170:
1157:
1152:
1137:
1136:
1121:
1120:
1105:
1104:
1086:
1085:
1073:
1072:
1053:
1051:
1050:
1045:
1043:
1042:
1026:
1024:
1023:
1018:
1002:
1000:
999:
994:
978:
976:
975:
970:
956:
955:
939:
937:
936:
931:
910:
908:
907:
902:
896:
895:
880:
879:
867:
866:
857:
856:
844:
843:
828:
827:
809:
808:
796:
795:
776:
774:
773:
768:
752:
750:
749:
744:
732:
730:
729:
724:
708:complex analysis
705:
703:
702:
697:
685:
683:
682:
677:
672:
667:
649:
647:
646:
641:
629:
627:
626:
621:
619:
618:
602:
600:
599:
594:
579:
577:
576:
571:
565:
564:
555:
554:
539:
538:
529:
528:
516:
515:
506:
505:
490:
489:
477:
476:
458:or equivalently
454:
452:
451:
446:
440:
439:
424:
423:
411:
410:
401:
400:
388:
387:
375:
374:
352:
350:
349:
344:
342:
341:
325:
323:
322:
317:
301:
299:
298:
293:
282:not a square in
281:
279:
278:
273:
252:
250:
249:
244:
238:
237:
222:
221:
209:
208:
199:
198:
186:
185:
173:
172:
153:
151:
150:
145:
129:
127:
126:
121:
105:
103:
102:
97:
2562:
2561:
2557:
2556:
2555:
2553:
2552:
2551:
2542:Elliptic curves
2532:
2531:
2496:
2489:
2486:Elliptic Curves
2482:
2479:
2478:
2446:
2434:
2433:
2429:
2378:
2377:
2373:
2321:
2320:
2313:
2261:
2260:
2256:
2189:
2188:
2184:
2175:
2173:
2171:
2147:
2146:
2142:
2137:
2115:
2091:
2090:
2062:
2061:
2042:
2041:
2022:
2017:
2016:
1988:
1987:
1968:
1967:
1948:
1943:
1942:
1923:
1922:
1899:
1898:
1897:be a field and
1879:
1878:
1875:
1851:
1850:
1847:
1823:
1822:
1799:
1798:
1795:
1765:
1760:
1759:
1738:
1720:
1715:
1710:
1709:
1659:
1624:
1623:
1601:
1600:
1581:
1576:
1575:
1544:
1543:
1522:
1517:
1516:
1497:
1496:
1465:
1464:
1445:
1444:
1425:
1424:
1405:
1404:
1381:
1380:
1377:
1336:
1311:
1310:
1308:field extension
1306:, but over the
1288:
1287:
1266:
1261:
1260:
1241:
1240:
1191:
1175:
1162:
1128:
1112:
1096:
1077:
1064:
1059:
1058:
1034:
1029:
1028:
1009:
1008:
1005:quadratic twist
985:
984:
947:
942:
941:
916:
915:
887:
871:
858:
848:
835:
819:
800:
787:
782:
781:
759:
758:
735:
734:
715:
714:
688:
687:
655:
654:
652:field extension
632:
631:
610:
605:
604:
585:
584:
556:
546:
530:
520:
507:
497:
481:
468:
463:
462:
431:
415:
402:
392:
379:
366:
358:
357:
333:
328:
327:
308:
307:
304:quadratic twist
284:
283:
258:
257:
229:
213:
200:
190:
177:
164:
159:
158:
136:
135:
112:
111:
88:
87:
84:
82:Quadratic twist
37:quadratic twist
17:
12:
11:
5:
2560:
2558:
2550:
2549:
2544:
2534:
2533:
2530:
2529:
2509:(4): 943–973.
2494:
2477:
2476:
2427:
2398:(4): 455–474.
2386:(2002-07-08).
2371:
2311:
2254:
2209:(3): 541–575.
2182:
2169:
2139:
2138:
2136:
2133:
2132:
2131:
2126:
2121:
2114:
2111:
2098:
2075:
2072:
2049:
2028:
2025:
2001:
1998:
1975:
1954:
1951:
1930:
1906:
1886:
1874:
1873:Generalization
1871:
1858:
1846:
1843:
1830:
1806:
1794:
1791:
1789:of the curve.
1772:
1768:
1745:
1741:
1737:
1734:
1727:
1723:
1718:
1697:
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1636:
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1608:
1587:
1584:
1563:
1560:
1557:
1554:
1551:
1542:. In fact, if
1529:
1525:
1504:
1484:
1481:
1478:
1475:
1472:
1452:
1443:there exist a
1432:
1412:
1388:
1376:
1373:
1360:
1357:
1354:
1351:
1348:
1343:
1339:
1335:
1331:
1327:
1324:
1321:
1318:
1295:
1273:
1269:
1248:
1237:
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1224:
1219:
1214:
1210:
1206:
1203:
1198:
1194:
1190:
1187:
1182:
1178:
1174:
1169:
1165:
1161:
1156:
1151:
1147:
1143:
1140:
1135:
1131:
1127:
1124:
1119:
1115:
1111:
1108:
1103:
1099:
1095:
1092:
1089:
1084:
1080:
1076:
1071:
1067:
1041:
1037:
1016:
992:
968:
965:
962:
959:
954:
950:
929:
926:
923:
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911:
899:
894:
890:
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865:
861:
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851:
847:
842:
838:
834:
831:
826:
822:
818:
815:
812:
807:
803:
799:
794:
790:
766:
755:elliptic curve
742:
722:
695:
675:
670:
665:
662:
639:
617:
613:
592:
581:
580:
568:
563:
559:
553:
549:
545:
542:
537:
533:
527:
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519:
514:
510:
504:
500:
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471:
456:
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340:
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315:
291:
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265:
254:
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241:
236:
232:
228:
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220:
216:
212:
207:
203:
197:
193:
189:
184:
180:
176:
171:
167:
143:
132:elliptic curve
119:
108:characteristic
106:is a field of
95:
83:
80:
57:quartic twists
29:elliptic curve
15:
13:
10:
9:
6:
4:
3:
2:
2559:
2548:
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2543:
2540:
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2522:
2517:
2512:
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2250:
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2222:
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2208:
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2200:
2196:
2192:
2186:
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2172:
2166:
2162:
2158:
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2144:
2141:
2134:
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2125:
2122:
2120:
2117:
2116:
2112:
2110:
2096:
2070:
2047:
2026:
2023:
1996:
1973:
1952:
1949:
1928:
1920:
1904:
1884:
1872:
1870:
1856:
1844:
1842:
1828:
1820:
1819:quartic twist
1804:
1793:Quartic twist
1792:
1790:
1788:
1770:
1766:
1743:
1739:
1735:
1732:
1725:
1721:
1716:
1695:
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1640:
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1585:
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1555:
1552:
1527:
1523:
1502:
1479:
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1402:
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1167:
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1129:
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1087:
1082:
1078:
1074:
1069:
1065:
1057:
1056:
1055:
1039:
1035:
1027:is the curve
1014:
1006:
990:
982:
966:
963:
960:
957:
952:
948:
927:
924:
921:
897:
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868:
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853:
849:
845:
840:
836:
832:
829:
824:
820:
816:
813:
810:
805:
801:
797:
792:
788:
780:
779:
778:
777:of the form:
764:
756:
740:
720:
711:
709:
693:
668:
660:
653:
637:
615:
611:
590:
566:
561:
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531:
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473:
469:
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441:
436:
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403:
397:
393:
389:
384:
380:
376:
371:
367:
363:
356:
355:
354:
338:
334:
326:is the curve
313:
305:
289:
269:
266:
263:
239:
234:
230:
226:
223:
218:
214:
210:
205:
201:
195:
191:
187:
182:
178:
174:
169:
165:
157:
156:
155:
154:of the form:
141:
133:
117:
109:
93:
86:First assume
81:
79:
77:
73:
69:
64:
62:
58:
54:
50:
46:
43:to E over an
42:
38:
34:
30:
26:
22:
2506:
2502:
2485:
2454:
2450:
2430:
2395:
2391:
2374:
2331:
2327:
2281:math/0508174
2271:
2267:
2257:
2206:
2202:
2185:
2174:. Retrieved
2151:
2143:
1876:
1848:
1818:
1796:
1621:
1401:finite field
1378:
1238:
1004:
913:
712:
706:, while the
582:
457:
303:
255:
85:
65:
56:
52:
36:
21:mathematical
18:
2457:(1): 1–23.
2380:Rubin, Karl
2334:: 474–490.
1845:Cubic twist
61:j-invariant
2536:Categories
2436:GouvĂŞa, F.
2341:1611.04856
2176:2022-04-10
2135:References
940:such that
41:isomorphic
23:field of
2440:Mazur, B.
2414:0273-0979
2366:119143097
2298:0012-7094
2241:0020-9910
2216:0904.3709
2195:Rubin, K.
2191:Mazur, B.
2074:¯
2000:¯
1736:−
925:∈
267:≠
31:E over a
2442:(1991).
2113:Examples
2027:′
1953:′
1758:, where
1586:′
2525:2152834
2471:2939253
2422:1920278
2346:Bibcode
2306:2326034
2249:3394387
2221:Bibcode
49:isogeny
19:In the
2523:
2469:
2420:
2412:
2364:
2304:
2296:
2247:
2239:
2167:
1003:, the
979:is an
914:Given
753:be an
302:, the
256:Given
130:be an
2521:JSTOR
2490:(PDF)
2467:JSTOR
2447:(PDF)
2362:S2CID
2336:arXiv
2302:S2CID
2276:arXiv
2274:(1).
2245:S2CID
2211:arXiv
1817:by a
1403:with
1399:is a
983:over
757:over
134:over
53:cubic
33:field
27:, an
2410:ISSN
2294:ISSN
2237:ISSN
2165:ISBN
2040:and
1259:and
603:and
55:and
2511:doi
2459:doi
2400:doi
2354:doi
2332:519
2286:doi
2272:137
2229:doi
2207:181
2157:doi
1966:of
1619:).
1515:or
1379:If
1007:of
306:of
2538::
2519:.
2505:.
2501:.
2465:.
2453:.
2449:.
2438:;
2418:MR
2416:.
2408:.
2396:39
2394:.
2390:.
2382:;
2360:.
2352:.
2344:.
2330:.
2326:.
2314:^
2300:.
2292:.
2284:.
2270:.
2266:.
2243:.
2235:.
2227:.
2219:.
2205:.
2201:.
2193:;
2163:.
2109:.
1371:.
78:.
63:.
2527:.
2513::
2507:8
2473:.
2461::
2455:4
2424:.
2402::
2368:.
2356::
2348::
2338::
2308:.
2288::
2278::
2251:.
2231::
2223::
2213::
2179:.
2159::
2097:K
2071:K
2048:C
2024:C
1997:K
1974:C
1950:C
1929:K
1905:C
1885:K
1857:K
1829:E
1805:E
1771:E
1767:t
1744:E
1740:t
1733:=
1726:d
1722:E
1717:t
1696:2
1693:+
1690:q
1687:2
1684:=
1680:|
1676:)
1673:K
1670:(
1665:d
1661:E
1656:|
1652:+
1648:|
1644:)
1641:K
1638:(
1635:E
1631:|
1607:2
1583:y
1562:)
1559:y
1556:,
1553:x
1550:(
1528:d
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1503:E
1483:)
1480:y
1477:,
1474:x
1471:(
1451:y
1431:x
1411:q
1387:K
1359:)
1356:d
1353:+
1350:X
1347:+
1342:2
1338:X
1334:(
1330:/
1326:]
1323:X
1320:[
1317:K
1294:K
1272:d
1268:E
1247:E
1223:.
1218:2
1213:3
1209:a
1205:d
1202:+
1197:6
1193:a
1189:+
1186:x
1181:4
1177:a
1173:+
1168:2
1164:x
1160:)
1155:2
1150:1
1146:a
1142:d
1139:+
1134:2
1130:a
1126:(
1123:+
1118:3
1114:x
1110:=
1107:y
1102:3
1098:a
1094:+
1091:y
1088:x
1083:1
1079:a
1075:+
1070:2
1066:y
1040:d
1036:E
1015:E
991:K
967:d
964:+
961:X
958:+
953:2
949:X
928:K
922:d
898:.
893:6
889:a
885:+
882:x
877:4
873:a
869:+
864:2
860:x
854:2
850:a
846:+
841:3
837:x
833:=
830:y
825:3
821:a
817:+
814:y
811:x
806:1
802:a
798:+
793:2
789:y
765:K
741:E
721:K
694:K
674:)
669:d
664:(
661:K
638:K
616:d
612:E
591:E
567:.
562:6
558:a
552:3
548:d
544:+
541:x
536:4
532:a
526:2
522:d
518:+
513:2
509:x
503:2
499:a
495:d
492:+
487:3
483:x
479:=
474:2
470:y
442:.
437:6
433:a
429:+
426:x
421:4
417:a
413:+
408:2
404:x
398:2
394:a
390:+
385:3
381:x
377:=
372:2
368:y
364:d
339:d
335:E
314:E
290:K
270:0
264:d
240:.
235:6
231:a
227:+
224:x
219:4
215:a
211:+
206:2
202:x
196:2
192:a
188:+
183:3
179:x
175:=
170:2
166:y
142:K
118:E
94:K
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