TABLE OF THE IMAGINARY ABELIAN

NUMBER FIELDS WITH RELATIVE CLASS

NUMBER ONE AND CLASS NUMBER > 1

This is the table of imaginary abelian number fields with relative class number one but class number > 1. There exist exactly 130 such fields: 10 fields are cyclic and the other 120 fields are noncyclic. The maximal conductor of these fields is 65689 = 13 · 31 · 163, which is the conductor of the biquadratic number field Q(\sqrt{−163}, \sqrt{−403}).

Description of the table. n, f and h^{+} denote the degree conductor, and the class number of the maximal real subfield, respectively. For the types of (the Galois group of) fields, ^{∗} means the oddness (imaginarity) of a generator of the character group associated with each field. We use the following notations in order to express generators of associated character groups. χ_{4} denotes the unique primitive Dirichlet character of conductor 4. ψ_{2^{m}} (m ≧ 3) denotes the even primitive Dirichlet character of conductor 2^{m} and order 2^{m−2} such that ψ_{q}(5) = e^{πi/2^{m−2}}. For odd prime numbers p, χ_{p} denotes the primitive Dirichlet character of conductor p and order p−1 such that χ_{p}(r) = e^{2πi/(p−1)}, and ψ_{p^{n}} (n ≧ 2) denotes the even primitive Dirichlet character of conductor p^{n} and order p^{n−1} such that ψ_{p^{n}}(r) = e^{2πi/p^{n−1}}, where r denotes the least positive primitive root modulo p^{n}.

Typeset by AMS-TEX 2 TABLE OF THE IMAGINARY ABELIAN FIELDS WITH h^{−} = 1, h^{+} > 1

Table

n Type Generators f Simple expression h^{+}
χ_{4}, χ_{4}ψ_{8}χ_{5}^{2} 40 Q(\sqrt{−1}, \sqrt{−10})  2
χ_{4}ψ_{8}, χ_{4}χ_{5}^{2} 40 Q(\sqrt{−2}, \sqrt{−5})  2
χ_{4}, χ_{3}χ_{5}^{2}  60 Q(\sqrt{−1}, \sqrt{−15})  2
χ_{3}, χ_{4}χ_{5}^{2}  60 Q(\sqrt{−3}, \sqrt{−5})  2
χ_{4}ψ_{8}, χ_{4}χ_{13}^{6} 104 Q(\sqrt{−2}, \sqrt{−13})  2
χ_{3}, χ_{5}^{2}χ_{7}^{3} 105 Q(\sqrt{−3}, \sqrt{−35})  2
χ_{7}^{3}, χ_{3}χ_{5}^{2} 105 Q(\sqrt{−7}, \sqrt{−15})  2
χ_{4}ψ_{8}, χ_{3}χ_{5}^{2} 120 Q(\sqrt{−2}, \sqrt{−15})  2
χ_{3}, χ_{4}ψ_{8}χ_{5}^{2} 120 Q(\sqrt{−3}, \sqrt{−10})  2
χ_{4}, χ_{5}^{2}χ_{7}^{3} 140 Q(\sqrt{−1}, \sqrt{−35})  2
χ_{4}χ_{5}^{2}, χ_{7}^{3} 140 Q(\sqrt{−5}, \sqrt{−7})  2
χ_{4}, χ_{4}ψ_{8}χ_{29}^{14} 232 Q(\sqrt{−1}, \sqrt{−58})  2
χ_{3}, ψ_{8}χ_{11}^{5} 264 Q(\sqrt{−3}, \sqrt{−22})  2
χ_{3}ψ_{8}, χ_{11}^{5} 264 Q(\sqrt{−6}, \sqrt{−11})  2
χ_{4}ψ_{8}, χ_{5}^{2}χ_{7}^{3} 280 Q(\sqrt{−2}, \sqrt{−35})  2
χ_{7}^{3}, χ_{4}ψ_{8}χ_{5}^{2} 280 Q(\sqrt{−7}, \sqrt{−10})  2
χ_{4}ψ_{8}, χ_{4}χ_{37}^{18} 296 Q(\sqrt{−2}, \sqrt{−37})  2
χ_{3}, χ_{5}^{2}χ_{23}^{11} 345 Q(\sqrt{−3}, \sqrt{−115}) 2
4 (2^{∗}, 2^{∗}) χ_{7}^{3}, χ_{3}χ_{17}^{8} 357 Q(\sqrt{−7}, \sqrt{−51})  2
χ_{4}, χ_{7}^{3}χ_{13}^{6} 364 Q(\sqrt{−1}, \sqrt{−91})  2
χ_{7}^{3}, χ_{4}χ_{13}^{6} 364 Q(\sqrt{−7}, \sqrt{−13})  2
χ_{3}ψ_{8}, χ_{19}^{9} 456 Q(\sqrt{−6}, \sqrt{−19})  2
χ_{4}, χ_{5}^{2}χ_{23}^{11} 460 Q(\sqrt{−1}, \sqrt{−115}) 2
χ_{7}^{3}, χ_{67}^{33} 469 Q(\sqrt{−7}, \sqrt{−67})  3
χ_{11}^{5}, χ_{43}^{21} 473 Q(\sqrt{−11}, \sqrt{−43}) 3
χ_{3}, χ_{11}^{5}χ_{17}^{8} 561 Q(\sqrt{−3}, \sqrt{−187}) 2
χ_{11}^{5}, χ_{3}χ_{17}^{8} 561 Q(\sqrt{−11}, \sqrt{−51}) 2
χ_{11}^{5}, χ_{4}χ_{13}^{6} 572 Q(\sqrt{−11}, \sqrt{−13}) 2
χ_{3}χ_{5}^{2}, χ_{43}^{21} 645 Q(\sqrt{−15}, \sqrt{−43}) 2
χ_{3}, χ_{4}ψ_{8}χ_{29}^{14} 696 Q(\sqrt{−3}, \sqrt{−58})  2
χ_{3}, χ_{5}^{2}χ_{47}^{23} 705 Q(\sqrt{−3}, \sqrt{−235}) 2
χ_{4}ψ_{8}, χ_{7}^{3}χ_{13}^{6} 728 Q(\sqrt{−2}, \sqrt{−91})  2
χ_{7}^{3}, χ_{5}^{2}χ_{23}^{11} 805 Q(\sqrt{−7}, \sqrt{−115}) 2
χ_{19}^{9}, χ_{43}^{21} 817 Q(\sqrt{−19}, \sqrt{−43}) 5
χ_{4}χ_{5}^{2}, χ_{43}^{21} 860 Q(\sqrt{−5}, \sqrt{−43})  2
χ_{7}^{3}, χ_{3}χ_{41}^{20} 861 Q(\sqrt{−7}, \sqrt{−123}) 2
χ_{4}ψ_{8}, χ_{5}^{2}χ_{23}^{11} 920 Q(\sqrt{−2}, \sqrt{−115}) 2

TABLE OF THE IMAGINARY ABELIAN FIELDS WITH h^{−} = 1, h^{+} > 1 3

Continued

n Type Generators  f Simple expression h^{+}
χ_{4}, χ_{5}^{2}χ_{47}^{23}  940 Q(\sqrt{−1}, \sqrt{−235}) 6
χ_{4}χ_{13}^{6}, χ_{19}^{9} 988 Q(\sqrt{−13}, \sqrt{−19}) 2
χ_{11}^{5}, χ_{7}^{3}χ_{13}^{6} 1001 Q(\sqrt{−11}, \sqrt{−91}) 2
χ_{3}χ_{5}^{2}, χ_{67}^{33} 1005 Q(\sqrt{−15}, \sqrt{−67}) 2
χ_{3}ψ_{8}, χ_{43}^{21} 1032 Q(\sqrt{−6}, \sqrt{−43})  2
χ_{4}ψ_{8}, χ_{163}^{81} 1304 Q(\sqrt{−2}, \sqrt{−163}) 3
χ_{7}^{3}, χ_{11}^{5}χ_{17}^{8} 1309 Q(\sqrt{−7}, \sqrt{−187}) 2
χ_{4}χ_{5}^{2}, χ_{67}^{33} 1340 Q(\sqrt{−5}, \sqrt{−67})  2
χ_{11}^{5}, χ_{3}χ_{41}^{20} 1353 Q(\sqrt{−11}, \sqrt{−123}) 2
χ_{5}^{2}χ_{7}^{3}, χ_{43}^{21} 1505 Q(\sqrt{−35}, \sqrt{−43}) 2
χ_{3}ψ_{8}, χ_{67}^{33} 1608 Q(\sqrt{−6}, \sqrt{−67})  2
χ_{4}, χ_{13}^{6}χ_{31}^{15} 1612 Q(\sqrt{−1}, \sqrt{−403}) 2
χ_{7}^{3}, χ_{5}^{2}χ_{47}^{23} 1645 Q(\sqrt{−7}, \sqrt{−235}) 2
χ_{19}^{9}, ψ_{8}χ_{11}^{5} 1672 Q(\sqrt{−19}, \sqrt{−22}) 2
χ_{4}, χ_{7}^{3}χ_{61}^{30} 1708 Q(\sqrt{−1}, \sqrt{−427}) 6
χ_{4}ψ_{8}χ_{5}^{2}, χ_{43}^{21} 1720 Q(\sqrt{−10}, \sqrt{−43}) 2
χ_{19}^{9}, χ_{7}^{3}χ_{13}^{6} 1729 Q(\sqrt{−19}, \sqrt{−91}) 2
χ_{4}ψ_{8}, χ_{5}^{2}χ_{47}^{23} 1880 Q(\sqrt{−2}, \sqrt{−235}) 2
χ_{7}^{3}, χ_{3}χ_{89}^{44} 1869 Q(\sqrt{−7}, \sqrt{−267}) 2
4 (2^{∗}, 2^{∗}) χ_{19}^{9}, χ_{3}χ_{41}^{20} 2337 Q(\sqrt{−19}, \sqrt{−123}) 2
χ_{5}^{2}χ_{7}^{3}, χ_{67}^{33} 2345 Q(\sqrt{−35}, \sqrt{−67}) 2
χ_{3}χ_{5}^{2}, χ_{163}^{81} 2445 Q(\sqrt{−15}, \sqrt{−163}) 2
χ_{11}^{5}, χ_{4}ψ_{8}χ_{29}^{14} 2552 Q(\sqrt{−11}, \sqrt{−58}) 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{67}^{33} 2680 Q(\sqrt{−10}, \sqrt{−67}) 2
χ_{19}^{9}, χ_{4}χ_{37}^{18} 2812 Q(\sqrt{−19}, \sqrt{−37}) 2
χ_{7}^{3}, χ_{13}^{6}χ_{31}^{15} 2821 Q(\sqrt{−7}, \sqrt{−403}) 2
χ_{4}ψ_{8}, χ_{13}^{6}χ_{31}^{15} 3224 Q(\sqrt{−2}, \sqrt{−403}) 2
χ_{4}χ_{5}^{2}, χ_{163}^{81} 3260 Q(\sqrt{−5}, \sqrt{−163}) 2
χ_{4}ψ_{8}, χ_{7}^{3}χ_{61}^{30} 3416 Q(\sqrt{−2}, \sqrt{−427}) 2
χ_{4}χ_{13}^{6}, χ_{67}^{33} 3484 Q(\sqrt{−13}, \sqrt{−67}) 2
ψ_{8}χ_{11}^{5}, χ_{43}^{21} 3784 Q(\sqrt{−22}, \sqrt{−43}) 2
χ_{3}ψ_{8}, χ_{163}^{81} 3912 Q(\sqrt{−6}, \sqrt{−163}) 2
χ_{19}^{9}, χ_{4}ψ_{8}χ_{29}^{14} 4408 Q(\sqrt{−19}, \sqrt{−58}) 2
χ_{11}^{5}, χ_{13}^{6}χ_{31}^{15} 4433 Q(\sqrt{−11}, \sqrt{−403}) 2
χ_{11}^{5}, χ_{7}^{3}χ_{61}^{30} 4697 Q(\sqrt{−11}, \sqrt{−427}) 2
χ_{43}^{21}, χ_{5}^{2}χ_{23}^{11} 4945 Q(\sqrt{−43}, \sqrt{−115}) 2
χ_{19}^{9}, χ_{3}χ_{89}^{44} 5073 Q(\sqrt{−19}, \sqrt{−267}) 2

4 TABLE OF THE IMAGINARY ABELIAN FIELDS WITH h^{−} = 1, h^{+} > 1

Continued

n Type Generators  f Simple expression h^{+}
χ_{5}^{2}χ_{7}^{3}, χ_{163}^{81}  5705 Q(\sqrt{−35}, \sqrt{−163}) 2
ψ_{8}χ_{11}^{5}, χ_{67}^{33} 5896 Q(\sqrt{−22}, \sqrt{−67})  2
χ_{67}^{33}, χ_{7}^{3}χ_{13}^{6}  6097 Q(\sqrt{−67}, \sqrt{−91}) 14
χ_{4}χ_{37}^{18}, χ_{43}^{21} 6364 Q(\sqrt{−37}, \sqrt{−43}) 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{163}^{81} 6520 Q(\sqrt{−10}, \sqrt{−163}) 2
χ_{19}^{9}, χ_{13}^{6}χ_{31}^{15} 7657 Q(\sqrt{−19}, \sqrt{−403}) 2
χ_{67}^{33}, χ_{5}^{2}χ_{23}^{11}  7705 Q(\sqrt{−67}, \sqrt{−115}) 10
χ_{67}^{33}, χ_{3}χ_{41}^{20}  8241 Q(\sqrt{−67}, \sqrt{−123}) 2
χ_{3}χ_{17}^{8}, χ_{163}^{81} 8313 Q(\sqrt{−51}, \sqrt{−163}) 2
χ_{4}χ_{13}^{6}, χ_{163}^{81} 8476 Q(\sqrt{−13}, \sqrt{−163}) 2
χ_{43}^{21}, χ_{4}ψ_{8}χ_{29}^{14} 9976 Q(\sqrt{−43}, \sqrt{−58})  2
χ_{43}^{21}, χ_{5}^{2}χ_{47}^{23} 10105 Q(\sqrt{−43}, \sqrt{−235}) 2
χ_{43}^{21}, χ_{3}χ_{89}^{44} 11481 Q(\sqrt{−43}, \sqrt{−267}) 2
4 (2^{∗}, 2^{∗}) ψ_{8}χ_{11}^{5}, χ_{163}^{81} 14344 Q(\sqrt{−22}, \sqrt{−163}) 2
χ_{7}^{3}χ_{13}^{6}, χ_{163}^{81} 14833 Q(\sqrt{−91}, \sqrt{−163}) 2
χ_{67}^{33}, χ_{5}^{2}χ_{47}^{23} 15745 Q(\sqrt{−67}, \sqrt{−235}) 2
χ_{43}^{21}, χ_{7}^{3}χ_{61}^{30} 18361 Q(\sqrt{−43}, \sqrt{−427}) 2
χ_{5}^{2}χ_{23}^{11}, χ_{163}^{81} 18745 Q(\sqrt{−115}, \sqrt{−163}) 2
χ_{4}χ_{37}^{18}, χ_{163}^{81} 24124 Q(\sqrt{−37}, \sqrt{−163}) 2
χ_{67}^{33}, χ_{13}^{6}χ_{31}^{15} 27001 Q(\sqrt{−67}, \sqrt{−403}) 2
χ_{67}^{33}, χ_{7}^{3}χ_{61}^{30} 28609 Q(\sqrt{−67}, \sqrt{−427}) 14
χ_{163}^{81}, χ_{11}^{5}χ_{17}^{8} 30481 Q(\sqrt{−163}, \sqrt{−187}) 2
χ_{4}ψ_{8}χ_{29}^{14}, χ_{163}^{81} 37816 Q(\sqrt{−58}, \sqrt{−163}) 2
χ_{163}^{81}, χ_{5}^{2}χ_{47}^{23} 38305 Q(\sqrt{−163}, \sqrt{−235}) 2
χ_{163}^{81}, χ_{3}χ_{89}^{44} 43521 Q(\sqrt{−163}, \sqrt{−267}) 2
χ_{163}^{81}, χ_{13}^{6}χ_{31}^{15} 65689 Q(\sqrt{−163}, \sqrt{−403}) 2
χ_{3}, χ_{7}^{2}ψ_{9}  63  3
χ_{3}, χ_{7}^{4}ψ_{9}  63  3
χ_{7}^{3}, χ_{7}^{4}ψ_{9}  63  3
χ_{7}^{3}, χ_{7}^{4}χ_{13}^{4}  91  3
6 (2^{∗}, 3) χ_{3}, ψ_{9}χ_{13}^{8}  117  3
χ_{7}^{3}, χ_{7}^{4}χ_{19}^{6}  133  3
χ_{19}^{9}, ψ_{9}χ_{19}^{12}  171  3
χ_{7}^{3}, χ_{7}^{2}χ_{31}^{10}  217  3
χ_{19}^{9}, χ_{13}^{4}χ_{19}^{10}  247  3

TABLE OF THE IMAGINARY ABELIAN FIELDS WITH h^{−} = 1, h^{+} > 1 5

Continued

n Type  Generators  f  Simple expression  h^{+}
(2^{∗}, 4)  χ_{4}, ψ_{16}χ_{5}^{2}  80 Q(\sqrt{−1}, \sqrt{5(2 + \sqrt{2})})  2
χ_{4}ψ_{16}, χ_{4}χ_{5}^{2}  80 Q(i sin(π/8), \sqrt{−5})  2
χ_{7}^{3}χ_{17}^{4}, χ_{3}  357 Q(\sqrt{−7(17 + 4\sqrt{17})}, \sqrt{−3}) 2 8 (4^{∗}, 2^{∗})  χ_{61}^{15}, χ_{7}^{3}  427 Q(\sqrt{−(61 + 6\sqrt{61})}, \sqrt{−7})  5
χ_{3}ψ_{16}, χ_{11}^{5}  528 Q(\sqrt{−3(2 + \sqrt{2})}, \sqrt{−11})  2
χ_{7}^{3}χ_{17}^{4}, χ_{11}^{5}  1309 Q(\sqrt{−7(17 + 4\sqrt{17})}, \sqrt{−11}) 2
(6^{∗}, 2^{∗})  χ_{7}, χ_{3}χ_{5}^{2}  105 Q(ζ_{7}, \sqrt{−15})  2
(4^{∗}, 3)  χ_{13}^{3}, χ_{7}^{2}χ_{13}^{4}  91  3
(2^{∗}, 2, 3) χ_{7}^{3}, χ_{13}^{6}, χ_{7}^{4}χ_{13}^{4}  91  3
χ_{3}, χ_{7}^{3}, χ_{7}^{4}ψ_{9}  63  3
χ_{4}ψ_{8}, χ_{4}χ_{13}^{6}, χ_{13}^{4} 104  2
χ_{3}, χ_{5}^{2}χ_{7}^{3}, χ_{7}^{2}  105 Q(\sqrt{−3}, \sqrt{−35}, cos(π/7))  2
χ_{4}, χ_{5}^{2}χ_{7}^{3}, χ_{7}^{2}  140 Q(\sqrt{−1}, \sqrt{−35}, cos(2π/7))  2 12 χ_{4}, χ_{3}χ_{5}^{2}, ψ_{9} 180 Q(\sqrt{−1}, \sqrt{−15}, cos(2π/9)) 2
(2^{∗}, 2^{∗}, 3) χ_{4}ψ_{8}, χ_{5}^{2}χ_{7}^{3}, χ_{7}^{2} 280 Q(\sqrt{−2}, \sqrt{−35}, cos(2π/7))  2
χ_{7}^{3}, χ_{3}χ_{5}^{2}, ψ_{9} 315 Q(\sqrt{−7}, \sqrt{−15}, cos(2π/9)) 2
χ_{7}^{3}, χ_{4}χ_{13}^{6}, χ_{13}^{4} 364  2
χ_{4}, χ_{3}χ_{5}^{2}, χ_{7}^{2} 420 Q(\sqrt{−1}, \sqrt{−15}, cos(2π/7)) 2
χ_{4}ψ_{8}, χ_{11}^{5}, χ_{7}^{2} 616 Q(\sqrt{−2}, \sqrt{−11}, cos(2π/7))  3
χ_{4}ψ_{8}, χ_{3}χ_{5}^{2}, χ_{7}^{2} 840 Q(\sqrt{−2}, \sqrt{−15}, cos(2π/7))  2
16 (8^{∗}, 2^{∗})  χ_{4}ψ_{32}, χ_{4}χ_{5}^{2}  160 Q(i sin(π/16), \sqrt{−5})  2