TABLE OF THE IMAGINARY BIBYCLIC BIQUADRATIC

NUMBER FIELDS WITH CLASS NUMBERS ≦ 50

M. Watkins completed the classification of the imaginary quadratic number fields with class numbers at most 100. Then we can determine the imaginary bibyclic biquadratic number fields with class numbers ≦ 50. We give here list of of such fields. We mean by (d_{1}, d_{2}) the field ℚ(\sqrt{d_{1}}, \sqrt{d_{2}}), by (d)_{H} the Hilbert class field of an imaginary quadratic number field (d) of discriminant d, and by (d)_{UQ} an unramified quadratic extension of (d).

References

M. Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), no. 246, 907– 938; MR2031415 (2005a:11175); Zbl 1050.11095.

Typeset by AMS-TEX

2 TABLE OF THE IMAGINARY BIQUADRATIC FIELDS WITH H ≦ 50

h = 1 (47 fields): 15 fields are Hilbert class fields of imaginary quadratic number fields with class number 2 and the other 32 fields are composita of two imaginary quadratic number fields with class number 1.

(−3, 5) = (−15)_{H}, (−4, 5) = (−20)_{H}, (−3, 8) = (−24)_{H}, (−7, 5) = (−35)_{H}, (−8, 5) = (−40)_{H}, (−3, 17) = (−51)_{H}, (−4, 13) = (−52)_{H}, (−11, 8) = (−88)_{H}, (−7, 13) = (−91)_{H}, (−3, 41) = (−123)_{H}, (−4, 37) = (−148)_{H}, (−11, 17) = (−187)_{H} (−8, 29) = (−232)_{H}, (−3, 89) = (−267)_{H}, (−7, 61) = (−427)_{H};

(−4, −8), (−4, −3), (−4, −7), (−4, −11), (−4, −19), (−4, −43), (−4, −67), (−4, −163), (−8, −3), (−8, −7), (−8, −11), (−8, −19), (−8, −43), (−8, −67), (−3, −7), (−3, −11), (−3, −19), (−3, −43), (−3, −67), (−3, −163), (−7, −11), (−7, −19), (−7, −43), (−7, −163), (−11, −19), (−11, −67), (−11, −163), (−19, −67), (−19, −163), (−43, −67), (−43, −163), (−67, −163).

h = 2 (160 fields): 20 fields are the unramified quadratic extension of an imaginary quadratic number field with class group C_{4}, 26 fields are an unramified quadratic extension of an imaginary quadratic number field with class group V_{4}, 94 fields are the composite of two imaginary quadratic number fields with class numbers 1 and 2, and the other 20 fields are the composite of two imaginary quadratic number fields with class number 2.

(−3, −39) = (−39)_{UQ}, (−11, −55) = (−55)_{UQ}, (−7, −56) = (−56)_{UQ}, (−4, −68) = (−68)_{UQ}, (−8, −136) = (−136)_{UQ}, (−7, −203) = (−203)_{UQ}, (−3, −219) = (−219)_{UQ}, (−7, −259) = (−259)_{UQ}, (−3, −291) = (−291)_{UQ}, (−4, −292) = (−292)_{UQ}, (−19, −323) = (−323)_{UQ}, (−8, −328) = (−328)_{UQ}, (−4, −388) = (−388)_{UQ}, (−3, −723) = (−723)_{UQ}, (−7, −763) = (−763)_{UQ}, (−4, −772) = (−772)_{UQ}, (−3, −1227) = (−1227)_{UQ}, (−11, −1243) = (−1243)_{UQ}, (−19, −1387) = (−1387)_{UQ}, (−11, −1507) = (−1507)_{UQ};

(−4, −84) = (−84)_{UQ}, (−3, −84) = (−84)_{UQ}, (−7, −84) = (−84)_{UQ}, (−4, −132) = (−132)_{UQ}, (−3, −132) = (−132)_{UQ}, (−7, −132) = (−132)_{UQ}, (−8, −168) = (−168)_{UQ}, (−3, −168) = (−168)_{UQ}, (−7, −168) = (−168)_{UQ}, (−4, −228) = (−228)_{UQ}, (−3, −228) = (−228)_{UQ}, (−19, −228) = (−228)_{UQ}, (−4, −372) = (−372)_{UQ}, (−3, −372) = (−372)_{UQ}, (−3, −483) = (−483)_{UQ}, (−7, −483) = (−483)_{UQ}, (−4, −532) = (−532)_{UQ}, (−7, −532) = (−532)_{UQ}, (−19, −532) = (−532)_{UQ}, (−3, −627) = (−627)_{UQ}, (−11, −627) = (−627)_{UQ}, (−19, −627) = (−627)_{UQ}, (−4, −708) = (−708)_{UQ}, (−3, −708) = (−708)_{UQ}, (−4, −1012) = (−1012)_{UQ}, (−11, −1012) = (−1012)_{UQ};

(−3, −20), (−3, −35), (−3, −40), (−3, −88), (−3, −115), (−3, −187), (−3, −232), (−3, −235), (−4, −15), (−4, −24), (−4, −35), (−4, −40), (−4, −88), (−4, −91), (−4, −115), (−4, −232), (−4, −403), (−7, −15), (−7, −20), (−7, −40), (−7, −51), (−7, −52), (−7, −115), (−7, −123), (−7, −187), (−7, −235), (−7, −267), (−7, −403), (−8, −15), (−8, −20), (−8, −24), (−8, −35), (−8, −52), (−8, −88), (−8, −91), (−8, −115), (−8, −148), (−8, −235), (−8, −403), (−8, −427), (−11, −24), (−11, −51), (−11, −52), (−11, −91), (−11, −123), (−11, −232), (−11, −403), (−11, −427), (−19, −24), (−19, −52), (−19, −88), (−19, −91), (−19, −123), (−19, −148), (−19, −232), (−19, −403), (−43, −15), (−43, −20), (−43, −24), (−43, −35), (−43, −40), (−43, −88), (−43, −115), (−43, −148), (−43, −232), (−43, −235), (−43, −267), (−43, −427), (−67, −15), (−67, −20), (−67, −24), (−67, −35), (−67, −40), (−67, −52), (−67, −88), (−67, −123), (−67, −235), (−67, −403), (−163, −15), (−163, −20), (−163, −24), (−163, −35), (−163, −40), (−163, −51), (−163, −52), (−163, −88), (−163, −91), (−163, −115), (−163, −148), (−163, −187), (−163, −232), (−163, −235), (−163, −267), (−163, −403);

(−15, −20), (−15, −35), (−15, −40), (−15, −115), (−15, −235), (−20, −35), (−20, −40), (−20, −115), (−20, −235), (−24, −88), (−35, −40), (−35, −115), (−35, −235), (−40, −115), (−40, −235), (−51, −187), (−52, −91), (−52, −403), (−91, −403), (−115, −235).

h = 3 (163 fields): 28 fields are the unramified quadratic extension of an imaginary quadratic number field with class group C_{6}, 2 fields are the Hilbert class field of an imaginary quadratic number field with class number 2, 3 fields are the composite of two imaginary quadratic number fields with class number 1, the other 130 fields are the composite of two imaginary quadratic number fields with class numbers 1 and 3.

(−3, −87) = (−87)_{UQ}, (−8, −104) = (−104)_{UQ}, (−4, −116) = (−116)_{UQ}, (−19, −152) = (−152)_{UQ}, (−4, −212) = (−212)_{UQ}, (−4, −244) = (−244)_{UQ}, (−19, −247) = (−247)_{UQ}, (−3, −339) = (−339)_{UQ}, (−3, −411) = (−411)_{UQ}, (−8, −424) = (−424)_{UQ}, (−4, −436) = (−436)_{UQ}, (−11, −451) = (−451)_{UQ}, (−4, −628) = (−628)_{UQ}, (−7, −707) = (−707)_{UQ}, (−8, −808) = (−808)_{UQ}, (−3, −843) = (−843)_{UQ}, (−3, −1059) = (−1059)_{UQ}, (−7, −1099) = (−1099)_{UQ}, (−4, −1108) = (−1108)_{UQ}, (−8, −1192) = (−1192)_{UQ}, (−7, −1267) = (−1267)_{UQ}, (−3, −1347) = (−1347)_{UQ}, (−3, −1563) = (−1563)_{UQ}, (−4, −1588) = (−1588)_{UQ}, (−19, −1843) = (−1843)_{UQ}, (−7, −2443) = (−2443)_{UQ}, (−11, −2563) = (−2563)_{UQ}, (−3, −2787) = (−2787)_{UQ};

(−23, −115) = (−115)_{H}, (−31, −403) = (−403)_{H};

(−8, −163), (−7, −67), (−11, −43);

(−3, −23), (−3, −31), (−3, −59), (−3, −83), (−3, −139), (−3, −211), (−3, −283), (−3, −307), (−3, −379), (−3, −499), (−3, −883), (−3, −907), (−4, −23), (−4, −31), (−4, −59), (−4, −83), (−4, −107), (−4, −139), (−4, −211), (−4, −283), (−4, −307), (−4, −331), (−4, −379), (−4, −547), (−4, −643), (−4, −883), (−4, −907), (−7, −23), (−7, −31), (−7, −59), (−7, −83), (−7, −107), (−7, −139), (−7, −211), (−7, −283), (−7, −307), (−7, −331), (−7, −379), (−7, −499), (−7, −547), (−7, −643), (−7, −883), (−7, −907), (−8, −23), (−8, −31), (−8, −59), (−8, −83), (−8, −107), (−8, −139), (−8, −211), (−8, −283), (−8, −307), (−8, −331), (−8, −379), (−8, −499), (−8, −547), (−8, −643), (−8, −907), (−11, −23), (−11, −31), (−11, −59), (−11, −83), (−11, −107), (−11, −139), (−11, −211), (−11, −283), (−11, −307), (−11, −331), (−11, −379), (−11, −499), (−11, −547), (−11, −643), (−11, −883), (−11, −907), (−19, −23), (−19, −31), (−19, −59), (−19, −83), (−19, −107), (−19, −139), (−19, −283), (−19, −307), (−19, −379), (−19, −499), (−19, −547), (−19, −643), (−19, −907), (−43, −23), (−43, −31), (−43, −59), (−43, −107), (−43, −139), (−43, −283), (−43, −307), (−43, −331), (−43, −379), (−43, −499), (−43, −547), (−43, −643), (−43, −883), (−43, −907), (−67, −23), (−67, −31), (−67, −59), (−67, −83), (−67, −107), (−67, −139), (−67, −211), (−67, −283), (−67, −307), (−67, −331), (−67, −379), (−67, −499), (−67, −547), (−67, −643), (−67, −883), (−67, −907), (−163, −23), (−163, −31), (−163, −59), (−163, −83), (−163, −107), (−163, −139), (−163, −211), (−163, −331), (−163, −379), (−163, −499), (−163, −547), (−163, −643), (−163, −883).

h = 4 (385 fields): 29 fields are the unramified quadratic extension of an imaginary quadratic number field with class group C_{8}, 66 fields are an unramified quadratic extension of an imaginary quadratic number field with class group C_{4} × C_{2}, 27 fields are an unramified quadratic extension of an imaginary quadratic number field with class group V_{4}, 45 fields are the composite of two imaginary quadratic number fields with class numbers 1 and 2, 19 fields are the composite of two imaginary quadratic number fields with class number 2, 142 fields are the composite of two imaginary quadratic number fields with class number 1 and class group C_{4}, 5 fields are the composite of two imaginary quadratic number fields with class number 1 and class group V_{4}, and 52 fields are the composite of two imaginary quadratic number fields with class number 2 and class group C_{4}.

(−19, −95) = (−95)_{UQ}, (−3, −111) = (−111)_{UQ}, (−4, −164) = (−164)_{UQ}, (−3, −183) = (−183)_{UQ}, (−7, −371) = (−371)_{UQ}, (−4, −452) = (−452)_{UQ}, (−4, −548) = (−548)_{UQ}, (−3, −579) = (−579)_{UQ}, (−11, −583) = (−583)_{UQ}, (−8, −712) = (−712)_{UQ}, (−8, −904) = (−904)_{UQ}, (−3, −939) = (−939)_{UQ}, (−11, −979) = (−979)_{UQ}, (−7, −1043) = (−1043)_{UQ}, (−4, −1252) = (−1252)_{UQ}, (−3, −1299) = (−1299)_{UQ}, (−4, −1348) = (−1348)_{UQ}, (−3, −1803) = (−1803)_{UQ}, (−4, −1828) = (−1828)_{UQ}, (−8, −1864) = (−1864)_{UQ}, (−7, −1939) = (−1939)_{UQ}, (−8, −2248) = (−2248)_{UQ}, (−3, −2307) = (−2307)_{UQ}, (−7, −2611) = (−2611)_{UQ}, (−7, −2947) = (−2947)_{UQ}, (−7, −3787) = (−2947)_{UQ}, (−11, −3883) = (−979)_{UQ}, (−3, −3969) = (−2307)_{UQ}, (−11, −5947) = (−979)_{UQ};

(−3, −264) = (−264)_{UQ}, (−8, −264) = (−264)_{UQ}, (−11, −264) = (−264)_{UQ}, (−3, −276) = (−276)_{UQ}, (−4, −276) = (−276)_{UQ}, (−4, −308) = (−308)_{UQ}, (−7, −308) = (−308)_{UQ}, (−11, −308) = (−308)_{UQ}, (−3, −456) = (−456)_{UQ}, (−8, −456) = (−456)_{UQ}, (−19, −456) = (−456)_{UQ}, (−3, −552) = (−552)_{UQ}, (−8, −552) = (−552)_{UQ}, (−3, −564) = (−564)_{UQ}, (−4, −564) = (−564)_{UQ}, (−7, −616) = (−616)_{UQ}, (−8, −616) = (−616)_{UQ}, (−11, −616) = (−616)_{UQ}, (−3, −651) = (−651)_{UQ}, (−7, −651) = (−651)_{UQ}, (−3, −852) = (−852)_{UQ}, (−4, −852) = (−852)_{UQ}, (−4, −868) = (−868)_{UQ}, (−7, −868) = (−868)_{UQ}, (−3, −987) = (−987)_{UQ}, (−7, −987) = (−987)_{UQ}, (−3, −1032) = (−1032)_{UQ}, (−8, −1032) = (−1032)_{UQ}, (−43, −1032) = (−1032)_{UQ}, (−3, −1128) = (−1128)_{UQ}, (−8, −1128) = (−1128)_{UQ}, (−4, −1204) = (−1204)_{UQ}, (−7, −1204) = (−1204)_{UQ}, (−43, −1204) = (−1204)_{UQ}, (−7, −1288) = (−1288)_{UQ}, (−8, −1288) = (−1288)_{UQ}. (−3, −1659) = (−1659)_{UQ}, (−7, −1659) = (−1659)_{UQ}, (−8, −1672) = (−1672)_{UQ}, (−11, −1672) = (−1672)_{UQ}, (−19, −1672) = (−1672)_{UQ}, (−7, −1771) = (−1771)_{UQ}, (−11, −1771) = (−1771)_{UQ}, (−3, −1947) = (−1947)_{UQ}, (−11, −1947) = (−1947)_{UQ}, (−3, −1992) = (−1992)_{UQ}, (−8, −1992) = (−1992)_{UQ}, (−3, −2139) = (−2139)_{UQ}, (−3, −2163) = (−2163)_{UQ}, (−7, −2163) = (−2163)_{UQ}, (−4, −2212) = (−2212)_{UQ}, (−19, −2451) = (−2451)_{UQ}, (−43, −2451) = (−2451)_{UQ}, (−7, −2632) = (−2632)_{UQ}, (−8, −2632) = (−2632)_{UQ}, (−3, −2667) = (−2667)_{UQ}, (−7, −2667) = (−2667)_{UQ}, (−3, −3243) = (−3243)_{UQ}, (−3, −3507) = (−3507)_{UQ}, (−7, −3507) = (−3507)_{UQ}, (−7, −4123) = (−4123)_{UQ}, (−19, −4123) = (−4123)_{UQ}, (−3, −4323) = (−4323)_{UQ}, (−11, −4323) = (−4323)_{UQ}, (−7, −5467) = (−5467)_{UQ}, (−11, −5467) = (−5467)_{UQ}.

(−3, −120) = (−120)_{UQ}, (−15, −120) = (−120)_{UQ}, (−24, −120) = (−120)_{UQ}, (−3, −195) = (−195)_{UQ}, (−15, −195) = (−195)_{UQ}, (−7, −280) = (−280)_{UQ}, (−35, −280) = (−280)_{UQ}, (−3, −312) = (−312)_{UQ}, (−24, −312) = (−312)_{UQ}, (−4, −340) = (−340)_{UQ}, (−20, −340) = (−340)_{UQ}, (−3, −408) = (−408)_{UQ}, (−24, −408) = (−408)_{UQ}, (−51, −408) = (−408)_{UQ}, (−15, −435) = (−435)_{UQ}, (−8, −520) = (−520)_{UQ}, (−40, −520) = (−520)_{UQ}, (−3, −555) = (−555)_{UQ}, (−15, −555) = (−555)_{UQ}, (−7, −595) = (−595)_{UQ}, (−35, −595) = (−595)_{UQ}, (−11, −715) = (−715)_{UQ}, (−19, −760) = (−760)_{UQ}, (−3, −795) = (−795)_{UQ}, (−15, −795) = (−795)_{UQ}, (−7, −1435) = (−1435)_{UQ}, (−35, −1435) = (−1435)_{UQ}.

(−3, −52), (−3, −91), (−3, −148), (−3, −403), (−3, −427), (−4, −51), (−4, −123), (−4, −187), (−4, −267), (−7, −24), (−7, −88), (−7, −148), (−7, −232), (−8, −51), (−8, −123), (−8, −187), (−8, −267), (−11, −15), (−11, −20), (−11, −35), (−11, −40), (−11, −115), (−11, −148), (−11, −235), (−11, −267), (−19, −15), (−19, −20), (−19, −35), (−19, −40), (−19, −51), (−19, −115), (−19, −187), (−19, −235), (−43, −51), (−43, −52), (−43, −91), (−43, −123), (−43, −187), (−43, −403), (−67, −51), (−67, −148), (−67, −187), (−67, −232), (−163, −123), (−163, −427).

(−15, −24), (−15, −51), (−15, −123), (−20, −52), (−20, −88), (−20, −148), (−24, −51), (−24, −52), (−24, −148), (−24, −267), (−24, −427), (−35, −91), (−35, −427), (−40, −88), (−51, −123), (−51, −267), (−88, −148), (−88, −187), (−123, −267).

(−4, −39), (−7, −39), (−8, −39), (−11, −39), (−19, −39), (−67, −39), (−163, −39), (−3, −55), (−4, −55), (−7, −55), (−8, −55), (−43, −55), (−67, −55), (−163, −55), (−3, −56), (−4, −56), (−8, −56), (−11, −56), (−19, −56), (−43, −56), (−67, −56), (−163, −56), (−3, −68), (−7, −68), (−11, −68), (−163, −68), (−3, −136), (−7, −136), (−11, −136), (−3, −155), (−4, −155), (−7, −155), (−8, −155), (−43, −155), (−67, −155), (−163, −155), (−3, −184), (−4, −184), (−8, −184), (−43, −184), (−67, −184), (−163, −184), (−3, −203), (−4, −203), (−8, −203), (−19, −203), (−43, −203), (−163, −203), (−7, −219), (−11, −219), (−43, −219), (−163, −219), (−4, −259), (−8, −259), (−19, −259), (−163, −259), (−7, −291), (−67, −291), (−7, −292), (−11, −292), (−43, −292), (−163, −292), (−3, −323), (−7, −323), (−11, −323), (−163, −323), (−3, −328), (−11, −328), (−19, −328), (−67, −328), (−3, −355), (−4, −355), (−7, −355), (−8, −355), (−67, −355), (−163, −355), (−7, −388), (−19, −388), (−3, −568), (−11, −568), (−19, −568), (−43, −568), (−67, −568), (−3, −667), (−4, −667), (−8, −667), (−11, −667), (−19, −667), (−43, −667), (−163, −667), (−7, −723), (−11, −723), (−19, −723), (−43, −723), (−163, −723), (−4, −763), (−8, −763), (−11, −763), (−19, −763), (−67, −763), (−163, −763), (−11, −772), (−3, −955), (−4, −955), (−8, −955), (−43, −955), (−163, −955), (−3, −1003), (−7, −1003), (−11, −1003), (−163, −1003), (−4, −1027), (−7, −1027), (−8, −1027), (−11, −1027), (−19, −1027), (−67, −1027), (−163, −1027), (−7, −1227), (−11, −1227), (−19, −1227), (−43, −1227), (−67, −1227), (−3, −1243), (−43, −1243), (−67, −1243), (−7, −1387), (−43, −1387), (−163, −1387), (−3, −1411), (−7, −1411), (−163, −1411), (−3, −1507), (−43, −1507), (−67, −1507), (−163, −1507), (−3, −1555), (−4, −1555), (−8, −1555), (−43, −1555), (−67, −1555), (−163, −1555).

(−4, −168), (−4, −120), (−4, −280), (−4, −312), (−4, −760).

(−52, −39), (−91, −39), (−403, −39), (−15, −55), (−20, −55), (−35, −55), (−40, −55), (−115, −55), (−235, −55), (−24, −56), (−88, −56), (−51, −68), (−187, −68), (−51, −136), (−187, −136), (−15, −155), (−20, −155), (−35, −155), (−40, −155), (−115, −155), (−235, −155), (−24, −184), (−88, −184), (−232, −203), (−148, −259), (−51, −323), (−187, −323), (−123, −328), (−15, −355), (−20, −355), (−35, −355), (−115, −355), (−235, −355), (−24, −568), (−88, −568), (−232, −667), (−15, −955), (−20, −955), (−40, −955), (−115, −955), (−235, −955), (−51, −1003), (−187, −1003), (−91, −1027), (−403, −1027), (−51, −1411), (−187, −1411), (−15, −1555), (−20, −1555), (−40, −1555), (−115, −1555), (−235, −1555).