TABLE OF THE IMAGINARY ABELIAN NUMBER FIELDS

WITH GENUS NUMBER EQUAL TO CLASS NUMBER > 1

This is the table of imaginary abelian number fields with genus number equal to class number > 1. There exist exactly such fields:

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 g = h > 1

Table

n Type Generators  f  Simple expression h^{−} h^{+} g
χ_{5}ψ_{8}  40 Q(\sqrt{−(5 + \sqrt{5})})  2 1 2
χ_{3}ψ_{16}  48 Q(\sqrt{−3(2 + \sqrt{2})})  2 1 2
χ_{3}χ_{4}χ_{5}  60 Q(\sqrt{−3(5 + \sqrt{5})/2})  4 1 4
χ_{5}χ_{13}^{6}  65 Q(\sqrt{−13(5 + \sqrt{5})/2})  2 1 2
χ_{5}^{2}χ_{13}^{3}  65 Q(\sqrt{−5(13 + 2\sqrt{13})})  2 1 2
χ_{4}χ_{5}^{2}ψ_{16}  80 Q(\sqrt{−5(2 + \sqrt{2})})  2 1 2
χ_{5}ψ_{16}  80 Q(\sqrt{−(10 + 3\sqrt{10})})  2 2 4
χ_{5}χ_{17}^{8}  85 Q(\sqrt{−17(5 + \sqrt{5})/2})  2 1 2 4^{∗} χ_{5}χ_{17}^{12}  85 Q(\sqrt{−5(85 + 9\sqrt{85})/2}) 2 2 4
ψ_{8}χ_{13}^{3}  104 Q(\sqrt{−2(13 + 2\sqrt{13})})  2 1 2
χ_{3}χ_{5}χ_{7}^{3}  105 Q(\sqrt{−21(5 + \sqrt{5})/2})  4 1 4
χ_{7}^{3}χ_{17}^{4}  119 Q(\sqrt{−7(17 + 4\sqrt{17})})  2 1 2
χ_{3}χ_{4}χ_{5}ψ_{8} 120 Q(\sqrt{−3(5 + \sqrt{5})})  4 1 4
χ_{4}χ_{5}χ_{7}^{3} 140 Q(\sqrt{−7(5 + \sqrt{5})/2}) 4 1 4
χ_{3}χ_{5}χ_{13}^{9} 195 Q(\sqrt{−3(65 + 8\sqrt{65})})  4 2 8
χ_{3}χ_{5}^{2}χ_{17}^{4} 255 Q(\sqrt{−15(17 + 4\sqrt{17})}) 4 1 4
4  χ_{4}ψ_{8}, χ_{3}ψ_{8} 24 Q(\sqrt{−2}, \sqrt{3})  2 1 2
χ_{4}, ψ_{8}χ_{5}^{2}  40 Q(\sqrt{−1}, \sqrt{10})  1 2 2
χ_{4}χ_{5}^{2}, ψ_{8}  40 Q(\sqrt{−5}, \sqrt{2})  2 1 2
χ_{4}χ_{5}^{2}, χ_{3}χ_{4} 60 Q(\sqrt{−5}, \sqrt{3})  2 1 2
χ_{3}, χ_{4}χ_{7}^{3}  84 Q(\sqrt{−3}, \sqrt{7})  2 1 2
χ_{4}, χ_{3}χ_{7}^{3}  84 Q(\sqrt{−1}, \sqrt{21})  2 1 2
χ_{7}^{3}, χ_{3}χ_{4}  84 Q(\sqrt{−7}, \sqrt{3})  2 1 2
χ_{4}ψ_{8}, χ_{4}χ_{11}^{5} 88 Q(\sqrt{−2}, \sqrt{11})  2 1 2
(2^{∗}, 2) χ_{3}χ_{5}^{2}, χ_{3}χ_{7}^{3} 105 Q(\sqrt{−15}, \sqrt{21})  2 1 2
χ_{4}ψ_{8}, χ_{4}χ_{3}χ_{5}^{2} 120 Q(\sqrt{−2}, \sqrt{15})  2 2 4
χ_{3}χ_{5}^{2}, χ_{3}χ_{4}ψ_{8} 120 Q(\sqrt{−15}, \sqrt{6})  2 1 2
χ_{4}χ_{5}^{2}, χ_{3}χ_{4}ψ_{8} 120 Q(\sqrt{−5}, \sqrt{6})  4 1 4
χ_{4}ψ_{8}χ_{5}^{2}, χ_{3}χ_{4} 120 Q(\sqrt{−10}, \sqrt{3})  4 1 4
χ_{3}, χ_{4}χ_{11}^{5} 132 Q(\sqrt{−3}, \sqrt{11})  2 1 2
χ_{4}, χ_{3}χ_{11}^{5} 132 Q(\sqrt{−1}, \sqrt{33})  2 1 2
χ_{11}^{5}, χ_{3}χ_{4} 132 Q(\sqrt{−11}, \sqrt{3})  2 1 2
χ_{4}χ_{5}^{2}, χ_{4}χ_{7}^{3} 140 Q(\sqrt{−5}, \sqrt{7})  2 1 2

TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1 3

Continued

n Type Generators  f Simple expression h^{−} h^{+} g
χ_{3}, χ_{4}ψ_{8}χ_{7}^{3} 168 Q(\sqrt{−3}, \sqrt{14})  2 1 2
χ_{4}, χ_{3}χ_{7}^{3}ψ_{8} 168 Q(\sqrt{−1}, \sqrt{42})  2 2 4
χ_{4}ψ_{8}, χ_{3}χ_{7}^{3} 168 Q(\sqrt{−2}, \sqrt{21})  2 1 2
χ_{7}^{3}, χ_{3}χ_{4}ψ_{8} 168 Q(\sqrt{−7}, \sqrt{6})  2 1 2
χ_{3}ψ_{8}, χ_{4}χ_{7}^{3} 168 Q(\sqrt{−6}, \sqrt{7})  4 1 4
χ_{3}, χ_{4}χ_{19}^{9} 228 Q(\sqrt{−3}, \sqrt{19})  2 1 2
χ_{4}, χ_{3}χ_{19}^{9} 228 Q(\sqrt{−1}, \sqrt{57})  2 1 2
χ_{19}^{9}, χ_{3}χ_{4} 228 Q(\sqrt{−19}, \sqrt{3})  2 1 2
χ_{3}ψ_{8}, χ_{4}ψ_{8}χ_{11}^{5} 264 Q(\sqrt{−6}, \sqrt{22})  4 1 4
χ_{3}ψ_{8}, χ_{3}χ_{11}^{5} 264 Q(\sqrt{−6}, \sqrt{33})  2 1 2
ψ_{8}χ_{11}^{5}, χ_{4}χ_{3}ψ_{8} 264 Q(\sqrt{−22}, \sqrt{6})  4 1 4
χ_{4}ψ_{8}, χ_{4}χ_{5}^{2}χ_{7}^{3} 280 Q(\sqrt{−2}, \sqrt{35})  2 2 4
χ_{4}χ_{5}^{2}, χ_{4}ψ_{8}χ_{7}^{3} 280 Q(\sqrt{−5}, \sqrt{14})  4 1 4
χ_{4}ψ_{8}χ_{5}^{2}, χ_{4}χ_{7}^{3} 280 Q(\sqrt{−10}, \sqrt{7})  4 1 4
χ_{4}ψ_{8}, χ_{3}χ_{4}χ_{13}^{6} 312 Q(\sqrt{−2}, \sqrt{39})  2 2 4
χ_{4}χ_{13}^{6}, χ_{4}χ_{3}ψ_{8} 312 Q(\sqrt{−13}, \sqrt{6})  4 1 4
χ_{3}χ_{5}^{2}, χ_{3}χ_{23}^{11} 345 Q(\sqrt{−15}, \sqrt{69})  2 1 2
4 (2^{∗}, 2) χ_{4}χ_{13}^{6}, χ_{4}χ_{7}^{3} 364 Q(\sqrt{−13}, \sqrt{7})  2 1 2
χ_{3}, χ_{4}χ_{31}^{15} 372 Q(\sqrt{−3}, \sqrt{31})  2 1 2
χ_{4}, χ_{3}χ_{31}^{15} 372 Q(\sqrt{−1}, \sqrt{93})  2 1 2
χ_{3}ψ_{8}, χ_{4}ψ_{8}χ_{19}^{9} 456 Q(\sqrt{−6}, \sqrt{38})  4 1 4
χ_{4}χ_{5}^{2}, χ_{4}χ_{23}^{11} 460 Q(\sqrt{−5}, \sqrt{23})  2 1 2
χ_{3}, χ_{7}^{3}χ_{23}^{11} 483 Q(\sqrt{−3}, \sqrt{161})  2 1 2
χ_{7}^{3}, χ_{3}χ_{23}^{11} 483 Q(\sqrt{−7}, \sqrt{69})  2 1 2
χ_{4}, χ_{7}^{3}χ_{19}^{9} 532 Q(\sqrt{−1}, \sqrt{133})  2 1 2
χ_{7}^{3}, χ_{4}χ_{19}^{9} 532 Q(\sqrt{−7}, \sqrt{19})  2 1 2
χ_{19}^{9}, χ_{4}χ_{7}^{3} 532 Q(\sqrt{−19}, \sqrt{7})  2 1 2
χ_{3}χ_{17}^{8}, χ_{3}χ_{11}^{5} 561 Q(\sqrt{−51}, \sqrt{33})  2 1 2
χ_{3}, χ_{11}^{5}χ_{19}^{9} 627 Q(\sqrt{−3}, \sqrt{209})  2 1 2
χ_{11}^{5}, χ_{3}χ_{19}^{9} 627 Q(\sqrt{−11}, \sqrt{57})  2 1 2
χ_{19}^{9}, χ_{3}χ_{11}^{5} 627 Q(\sqrt{−19}, \sqrt{33})  2 1 2
χ_{3}ψ_{8}, ψ_{8}χ_{29}^{14} 696 Q(\sqrt{−6}, \sqrt{58})  2 2 4
χ_{3}χ_{5}^{2}, χ_{3}χ_{47}^{23} 705 Q(\sqrt{−15}, \sqrt{141})  2 1 2
χ_{3}, χ_{4}χ_{59}^{29} 708 Q(\sqrt{−3}, \sqrt{59})  2 1 2
χ_{4}, χ_{3}χ_{59}^{29} 708 Q(\sqrt{−1}, \sqrt{177})  2 1 2

4 TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1

Continued

n Type Generators f Simple expression h^{−} h^{+} g
χ_{3}ψ_{8}, χ_{4}ψ_{8}χ_{31}^{15} 744 Q(\sqrt{−6}, \sqrt{62})  4 1 4
χ_{4}ψ_{8}, χ_{4}χ_{5}^{2}χ_{19}^{9} 760 Q(\sqrt{−2}, \sqrt{95})  2 2 4
χ_{4}χ_{5}^{2}, χ_{4}ψ_{8}χ_{19}^{9} 760 Q(\sqrt{−5}, \sqrt{38})  4 1 4
χ_{4}ψ_{8}χ_{5}^{2}, χ_{4}χ_{19}^{9} 760 Q(\sqrt{−10}, \sqrt{19})  4 1 4
χ_{4}χ_{5}^{2}, χ_{4}χ_{47}^{23} 940 Q(\sqrt{−5}, \sqrt{47}) 2 1 2 (2^{∗}, 2) χ_{11}^{5}, χ_{4}χ_{23}^{11} 1012 Q(\sqrt{−11}, \sqrt{23})  2 1 2
χ_{4}χ_{13}^{6}, χ_{4}χ_{31}^{15} 1612 Q(\sqrt{−13}, \sqrt{31})  2 1 2
ψ_{8}χ_{11}^{5}, χ_{4}ψ_{8}χ_{23}^{11} 2024 Q(\sqrt{−22}, \sqrt{46})  4 1 4
χ_{7}^{3}χ_{13}^{6}, χ_{7}^{3}χ_{31}^{15} 2821 Q(\sqrt{−91}, \sqrt{217})  2 1 2
χ_{4}, ψ_{8}χ_{3}  24 Q(\sqrt{−1}, \sqrt{−6})  2 1 2
χ_{4}ψ_{8}, χ_{4}χ_{5}^{2}  40 Q(\sqrt{−2}, \sqrt{−5})  1 2 2
χ_{4}, χ_{3}χ_{5}^{2}  60 Q(\sqrt{−1}, \sqrt{−15})  1 2 2
χ_{3}, χ_{4}χ_{5}^{2}  60 Q(\sqrt{−3}, \sqrt{−5})  1 2 2
χ_{4}, ψ_{8}χ_{11}^{5}  88 Q(\sqrt{−1}, \sqrt{−22})  2 1 2
χ_{4}ψ_{8}, χ_{4}χ_{13}^{6}  104 Q(\sqrt{−2}, \sqrt{−13})  1 2 2
χ_{3}, χ_{5}^{2}χ_{7}^{3}  105 Q(\sqrt{−3}, \sqrt{−35})  1 2 2
χ_{7}^{3}, χ_{3}χ_{5}^{2}  105 Q(\sqrt{−7}, \sqrt{−15})  1 2 2
χ_{3}, χ_{4}ψ_{8}χ_{5}^{2}  120 Q(\sqrt{−3}, \sqrt{−10})  1 2 2 4 χ_{4}, χ_{3}χ_{5}^{2}ψ_{8}  120 Q(\sqrt{−1}, \sqrt{−30})  2 2 4
χ_{4}ψ_{8}, χ_{3}χ_{5}^{2}  120 Q(\sqrt{−2}, \sqrt{−15})  1 2 2
χ_{4}, χ_{5}^{2}χ_{7}^{3}  140 Q(\sqrt{−1}, \sqrt{−35})  1 2 2
χ_{4}χ_{5}^{2}, χ_{7}^{3}  140 Q(\sqrt{−5}, \sqrt{−7})  1 2 2
χ_{4}ψ_{8}, χ_{4}χ_{3}χ_{7}^{3} 168 Q(\sqrt{−2}, \sqrt{−21})  2 2 4 (2^{∗}, 2^{∗}) χ_{3}ψ_{8}, χ_{3}χ_{4}χ_{7}^{3} 168 Q(\sqrt{−6}, \sqrt{−21})  4 1 4
χ_{4}, χ_{4}ψ_{8}χ_{29}^{14}  232 Q(\sqrt{−1}, \sqrt{−58})  1 2 2
χ_{3}, ψ_{8}χ_{11}^{5}  264 Q(\sqrt{−3}, \sqrt{−22})  1 2 2
χ_{3}ψ_{8}, χ_{11}^{5}  264 Q(\sqrt{−6}, \sqrt{−11})  1 2 2
χ_{4}, χ_{5}^{2}χ_{7}^{3}ψ_{8}  280 Q(\sqrt{−1}, \sqrt{−70})  2 2 4
χ_{4}ψ_{8}, χ_{5}^{2}χ_{7}^{3}  280 Q(\sqrt{−2}, \sqrt{−35})  1 2 2
χ_{7}^{3}, χ_{4}ψ_{8}χ_{5}^{2}  280 Q(\sqrt{−7}, \sqrt{−10})  1 2 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{5}^{2}χ_{7}^{3} 280 Q(\sqrt{−10}, \sqrt{−35}) 2 1 2
χ_{4}ψ_{8}, χ_{4}χ_{37}^{18}  296 Q(\sqrt{−2}, \sqrt{−37})  1 2 2
χ_{4}, χ_{3}ψ_{8}χ_{13}^{6}  312 Q(\sqrt{−1}, \sqrt{−78})  2 2 4
χ_{3}ψ_{8}, χ_{4}χ_{13}^{6}  312 Q(\sqrt{−6}, \sqrt{−13})  2 2 4
χ_{3}, χ_{5}^{2}χ_{23}^{11}  345 Q(\sqrt{−3}, \sqrt{−115}) 1 2 2
χ_{7}^{3}, χ_{3}χ_{17}^{8}  357 Q(\sqrt{−7}, \sqrt{−51})  1 2 2

TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1 5

Continued

n Type Generators  f Simple expression h^{−} h^{+} g
χ_{4}, χ_{7}^{3}χ_{13}^{6}  364 Q(\sqrt{−1}, \sqrt{−91})  1 2 2
χ_{7}^{3}, χ_{4}χ_{13}^{6}  364 Q(\sqrt{−7}, \sqrt{−13})  1 2 2
χ_{4}χ_{5}^{2}, ψ_{8}χ_{11}^{5} 440 Q(\sqrt{−5}, \sqrt{−22})  2 2 4
χ_{4}ψ_{8}χ_{5}^{2}, ψ_{8}χ_{11}^{5} 440 Q(\sqrt{−10}, \sqrt{−22}) 2 2 4
χ_{3}ψ_{8}, χ_{19}^{9} 456 Q(\sqrt{−6}, \sqrt{−19}) 1 2 2
χ_{4}, χ_{5}^{2}χ_{23}^{11}  460 Q(\sqrt{−1}, \sqrt{−115}) 1 2 2
χ_{11}^{5}, χ_{3}χ_{17}^{8}  561 Q(\sqrt{−11}, \sqrt{−51}) 1 2 2
χ_{11}^{5}, χ_{4}χ_{13}^{6}  572 Q(\sqrt{−11}, \sqrt{−13}) 1 2 2
χ_{3}χ_{5}^{2}, χ_{43}^{21}  645 Q(\sqrt{−15}, \sqrt{−43}) 1 2 2
χ_{3}, χ_{4}ψ_{8}χ_{29}^{14} 696 Q(\sqrt{−3}, \sqrt{−58})  1 2 2
χ_{3}, χ_{5}^{2}χ_{47}^{23}  705 Q(\sqrt{−3}, \sqrt{−235}) 1 2 2
χ_{4}ψ_{8}, χ_{7}^{3}χ_{13}^{6} 728 Q(\sqrt{−2}, \sqrt{−91})  1 2 2
χ_{4}ψ_{8}, χ_{3}χ_{4}χ_{31}^{15} 744 Q(\sqrt{−2}, \sqrt{−93})  2 2 4
χ_{4}, χ_{5}^{2}ψ_{8}χ_{19}^{9} 760 Q(\sqrt{−1}, \sqrt{−190}) 2 2 4
χ_{7}^{3}, χ_{5}^{2}χ_{23}^{11}  805 Q(\sqrt{−7}, \sqrt{−115}) 1 2 2
χ_{5}^{2}χ_{7}^{3}, χ_{5}^{2}χ_{23}^{11} 805 Q(\sqrt{−35}, \sqrt{−115}) 2 1 2
χ_{4}χ_{5}^{2}, χ_{43}^{21} 860 Q(\sqrt{−5}, \sqrt{−43}) 1 2 2
χ_{7}^{3}, χ_{3}χ_{41}^{20}  861 Q(\sqrt{−7}, \sqrt{−123}) 1 2 2 4 (2^{∗}, 2^{∗}) χ_{3}ψ_{8}, χ_{4}χ_{37}^{18} 888 Q(\sqrt{−6}, \sqrt{−37})  2 2 4
χ_{4}ψ_{8}, χ_{5}^{2}χ_{23}^{11} 920 Q(\sqrt{−2}, \sqrt{−115}) 1 2 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{5}^{2}χ_{23}^{11} 920 Q(\sqrt{−10}, \sqrt{−115}) 2 1 2
χ_{4}χ_{13}^{6}, χ_{19}^{9}  988 Q(\sqrt{−13}, \sqrt{−19}) 1 2 2
χ_{11}^{5}, χ_{7}^{3}χ_{13}^{6} 1001 Q(\sqrt{−11}, \sqrt{−91}) 1 2 2
χ_{3}χ_{5}^{2}, χ_{67}^{33} 1005 Q(\sqrt{−15}, \sqrt{−67}) 1 2 2
χ_{3}ψ_{8}, χ_{43}^{21} 1032 Q(\sqrt{−6}, \sqrt{−43})  1 2 2
χ_{4}ψ_{8}, χ_{7}^{3}χ_{19}^{9} 1064 Q(\sqrt{−2}, \sqrt{−133}) 2 2 4
χ_{7}^{3}, χ_{11}^{5}χ_{17}^{8} 1309 Q(\sqrt{−7}, \sqrt{−187}) 1 2 2
χ_{4}χ_{5}^{2}, χ_{67}^{33} 1340 Q(\sqrt{−5}, \sqrt{−67})  1 2 2
χ_{11}^{5}, χ_{3}χ_{41}^{20} 1353 Q(\sqrt{−11}, \sqrt{−123}) 1 2 2
χ_{5}^{2}χ_{7}^{3}, χ_{43}^{21} 1505 Q(\sqrt{−35}, \sqrt{−43}) 1 2 2
χ_{3}ψ_{8}, χ_{67}^{33} 1608 Q(\sqrt{−6}, \sqrt{−67})  1 2 2
χ_{4}, χ_{13}^{6}χ_{31}^{15} 1612 Q(\sqrt{−1}, \sqrt{−403}) 1 2 2
χ_{7}^{3}, χ_{5}^{2}χ_{47}^{23} 1645 Q(\sqrt{−7}, \sqrt{−235}) 1 2 2
χ_{5}^{2}χ_{7}^{3}, χ_{5}^{2}χ_{47}^{23} 1645 Q(\sqrt{−35}, \sqrt{−235}) 2 1 2
χ_{19}^{9}, ψ_{8}χ_{11}^{5} 1672 Q(\sqrt{−19}, \sqrt{−22}) 1 2 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{43}^{21} 1720 Q(\sqrt{−10}, \sqrt{−43}) 1 2 2

6 TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1

Continued

n Type Generators  f Simple expression h^{−} h^{+} g
χ_{19}^{9}, χ_{7}^{3}χ_{13}^{6}  1729 Q(\sqrt{−19}, \sqrt{−91})  1 2 2
χ_{7}^{3}, χ_{3}χ_{89}^{44}  1869 Q(\sqrt{−7}, \sqrt{−267})  1 2 2
χ_{4}ψ_{8}, χ_{5}^{2}χ_{47}^{23} 1880 Q(\sqrt{−2}, \sqrt{−235})  1 2 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{5}^{2}χ_{47}^{23} 1880 Q(\sqrt{−10}, \sqrt{−235}) 2 1 2
χ_{19}^{9}, χ_{3}χ_{41}^{20}  2337 Q(\sqrt{−19}, \sqrt{−123}) 1 2 2
χ_{5}^{2}χ_{7}^{3}, χ_{67}^{33} 2345 Q(\sqrt{−35}, \sqrt{−67}) 1 2 2
χ_{3}χ_{5}^{2}, χ_{163}^{81}  2445 Q(\sqrt{−15}, \sqrt{−163}) 1 2 2
χ_{11}^{5}, χ_{4}ψ_{8}χ_{29}^{14} 2552 Q(\sqrt{−11}, \sqrt{−58})  1 2 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{67}^{33} 2680 Q(\sqrt{−10}, \sqrt{−67})  1 2 2
χ_{19}^{9}, χ_{4}χ_{37}^{18}  2812 Q(\sqrt{−19}, \sqrt{−37})  1 2 2
χ_{7}^{3}, χ_{13}^{6}χ_{31}^{15}  2821 Q(\sqrt{−7}, \sqrt{−403})  1 2 2
χ_{4}ψ_{8}, χ_{13}^{6}χ_{31}^{15} 3224 Q(\sqrt{−2}, \sqrt{−403})  1 2 2
ψ_{8}χ_{11}^{5}, χ_{4}χ_{37}^{18} 3256 Q(\sqrt{−22}, \sqrt{−37})  2 2 4
χ_{4}χ_{5}^{2}, χ_{163}^{81} 3260 Q(\sqrt{−5}, \sqrt{−163}) 1 2 2
χ_{4}ψ_{8}, χ_{7}^{3}χ_{61}^{30} 3416 Q(\sqrt{−2}, \sqrt{−427})  1 2 2
χ_{4}χ_{13}^{6}, χ_{67}^{33}  3484 Q(\sqrt{−13}, \sqrt{−67})  1 2 2
ψ_{8}χ_{11}^{5}, χ_{43}^{21}  3784 Q(\sqrt{−22}, \sqrt{−43})  1 2 2
χ_{3}ψ_{8}, χ_{163}^{81}  3912 Q(\sqrt{−6}, \sqrt{−163})  1 2 2 4 (2^{∗}, 2^{∗}) χ_{19}^{9}, χ_{4}ψ_{8}χ_{29}^{14} 4408 Q(\sqrt{−19}, \sqrt{−58})  1 2 2
χ_{11}^{5}, χ_{13}^{6}χ_{31}^{15} 4433 Q(\sqrt{−11}, \sqrt{−403}) 1 2 2
χ_{11}^{5}, χ_{7}^{3}χ_{61}^{30}  4697 Q(\sqrt{−11}, \sqrt{−427}) 1 2 2
χ_{43}^{21}, χ_{5}^{2}χ_{23}^{11}  4945 Q(\sqrt{−43}, \sqrt{−115}) 1 2 2
χ_{19}^{9}, χ_{3}χ_{89}^{44}  5073 Q(\sqrt{−19}, \sqrt{−267}) 1 2 2
χ_{5}^{2}χ_{23}^{11}, χ_{5}^{2}χ_{47}^{23} 5405 Q(\sqrt{−115}, \sqrt{−235}) 2 1 2
χ_{5}^{2}χ_{7}^{3}, χ_{163}^{81}  5705 Q(\sqrt{−35}, \sqrt{−163}) 1 2 2
ψ_{8}χ_{11}^{5}, χ_{67}^{33}  5896 Q(\sqrt{−22}, \sqrt{−67})  1 2 2
χ_{4}χ_{37}^{18}, χ_{43}^{21}  6364 Q(\sqrt{−37}, \sqrt{−43})  1 2 2
χ_{4}ψ_{8}χ_{5}^{2}, χ_{163}^{81} 6520 Q(\sqrt{−10}, \sqrt{−163}) 1 2 2
χ_{19}^{9}, χ_{13}^{6}χ_{31}^{15} 7657 Q(\sqrt{−19}, \sqrt{−403}) 1 2 2
χ_{67}^{33}, χ_{3}χ_{41}^{20}  8241 Q(\sqrt{−67}, \sqrt{−123}) 1 2 2
χ_{3}χ_{17}^{8}, χ_{163}^{81} 8313 Q(\sqrt{−51}, \sqrt{−163}) 1 2 2
χ_{4}χ_{13}^{6}, χ_{163}^{81} 8476 Q(\sqrt{−13}, \sqrt{−163}) 1 2 2
χ_{43}^{21}, χ_{4}ψ_{8}χ_{29}^{14} 9976 Q(\sqrt{−43}, \sqrt{−58})  1 2 2
χ_{43}^{21}, χ_{5}^{2}χ_{47}^{23} 10105 Q(\sqrt{−43}, \sqrt{−235}) 1 2 2
χ_{43}^{21}, χ_{3}χ_{89}^{44} 11481 Q(\sqrt{−43}, \sqrt{−267}) 1 2 2
ψ_{8}χ_{11}^{5}, χ_{163}^{81} 14344 Q(\sqrt{−22}, \sqrt{−163}) 1 2 2

TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1 7

Continued

n Type Generators  f Simple expression h^{−} h^{+} g
χ_{7}^{3}χ_{13}^{6}, χ_{163}^{81} 14833 Q(\sqrt{−91}, \sqrt{−163})  1 2 2
χ_{67}^{33}, χ_{5}^{2}χ_{47}^{23} 15745 Q(\sqrt{−67}, \sqrt{−235})  1 2 2
χ_{43}^{21}, χ_{7}^{3}χ_{61}^{30} 18361 Q(\sqrt{−43}, \sqrt{−427})  1 2 2
χ_{5}^{2}χ_{23}^{11}, χ_{163}^{81} 18745 Q(\sqrt{−115}, \sqrt{−163})  1 2 2
χ_{4}χ_{37}^{18}, χ_{163}^{81} 24124 Q(\sqrt{−37}, \sqrt{−163})  1 2 2
4 (2^{∗}, 2^{∗}) χ_{67}^{33}, χ_{13}^{6}χ_{31}^{15} 27001 Q(\sqrt{−67}, \sqrt{−403})  1 2 2
χ_{163}^{81}, χ_{11}^{5}χ_{17}^{8} 30481 Q(\sqrt{−163}, \sqrt{−187})  1 2 2
χ_{4}ψ_{8}χ_{29}^{14}, χ_{163}^{81} 37816 Q(\sqrt{−58}, \sqrt{−163})  1 2 2
χ_{163}^{81}, χ_{5}^{2}χ_{47}^{23} 38305 Q(\sqrt{−163}, \sqrt{−235})  1 2 2
χ_{163}^{81}, χ_{3}χ_{89}^{44} 43521 Q(\sqrt{−163}, \sqrt{−267})  1 2 2
χ_{163}^{81}, χ_{13}^{6}χ_{31}^{15} 65689 Q(\sqrt{−163}, \sqrt{−403})  1 2 2
χ_{5}^{2}χ_{7}^{3}, χ_{7}^{2}  35 Q(\sqrt{−35}, cos(2π/7)) 2 1 2
χ_{3}χ_{5}^{2}, ψ_{9}  45 Q(\sqrt{−15}, cos(2π/9)) 2 1 2
χ_{4}χ_{13}^{6}, χ_{13}^{4}  52  2 1 2
χ_{3}, χ_{7}^{2}ψ_{9}  63  1 3 3
χ_{3}, χ_{7}^{4}ψ_{9}  63  1 3 3
χ_{7}^{3}, χ_{7}^{4}ψ_{9}  63  1 3 3
χ_{3}ψ_{8}, ψ_{9}  72 Q(\sqrt{−6}, cos(2π/9)) 2 1 2
χ_{3}χ_{4}χ_{7}^{3}, χ_{7}^{2}  84 Q(\sqrt{−21}, cos(2π/7)) 4 1 4
6 (2^{∗}, 3) χ_{7}^{3}, χ_{7}^{4}χ_{13}^{4}  91  1 3 3
χ_{7}^{3}χ_{13}^{6}, χ_{7}^{2}χ_{13}^{4}  91  2 3 6
χ_{7}^{3}χ_{13}^{6}, χ_{7}^{4}χ_{13}^{4}  91  2 3 6
χ_{3}χ_{5}^{2}, χ_{7}^{2}  105 Q(\sqrt{−15}, cos(2π/7)) 2 1 2
χ_{3}, ψ_{9}χ_{13}^{8}  117  1 3 3
χ_{7}^{3}, χ_{7}^{4}χ_{19}^{6}  133  1 3 3
χ_{19}^{9}, ψ_{9}χ_{19}^{12}  171  1 3 3
χ_{7}^{3}, χ_{7}^{2}χ_{31}^{10}  217  1 3 3
χ_{19}^{9}, χ_{13}^{4}χ_{19}^{10}  247  1 3 3
χ_{3}χ_{17}^{2}  51  2 1 2 8 8^{∗} χ_{5}χ_{17}^{2}  85  2 2 4

8 TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1

Continued

n Type Generators  f  Simple expression  h^{−} h^{+} g
χ_{3}, χ_{4}χ_{5}  60 Q(\sqrt{−3}, \sqrt{(5 + \sqrt{5})/2})  2 1 2
χ_{4}, χ_{3}χ_{5}  60 Q(\sqrt{−1}, \sqrt{3(5 + \sqrt{5})/2})  2 1 2
χ_{4}, ψ_{16}χ_{5}^{2}  80 Q(\sqrt{−1}, \sqrt{5(2 + \sqrt{2})})  1 2 2
χ_{4}χ_{5}^{2}, ψ_{16}  80 Q(\sqrt{−5}, cos(π/8))  2 1 2
χ_{3}, χ_{5}χ_{7}^{3}  105 Q(\sqrt{−3}, \sqrt{7(5 + \sqrt{5})/2})  2 1 2
(2^{∗}, 4)  χ_{7}^{3}, χ_{3}χ_{5}  105 Q(\sqrt{−7}, \sqrt{3(5 + \sqrt{5})/2})  2 1 2
χ_{3}, χ_{4}χ_{5}ψ_{8} 120 Q(\sqrt{−3}, \sqrt{(5 + \sqrt{5})})  2 1 2
χ_{4}, χ_{3}χ_{5}ψ_{8} 120 Q(\sqrt{−1}, \sqrt{3(5 + \sqrt{5})})  2 2 4
χ_{4}ψ_{8}, χ_{3}χ_{5} 120 Q(\sqrt{−2}, \sqrt{3(5 + \sqrt{5})/2})  2 2 4
χ_{4}, χ_{5}χ_{7}^{3}  140 Q(\sqrt{−1}, \sqrt{7(5 + \sqrt{5})/2})  2 1 2
χ_{7}^{3}, χ_{4}χ_{5}  140 Q(\sqrt{−7}, \sqrt{(5 + \sqrt{5})/2})  2 1 2
χ_{5}, χ_{3}χ_{4}  48 Q(i sin(π/8), \sqrt{3})  2 1 2
χ_{5}, χ_{3}χ_{4}  60 Q(ζ_{5}, \sqrt{3})  2 1 2
χ_{5}, χ_{3}χ_{7}^{3}  105 Q(ζ_{5}, \sqrt{21})  2 1 2
χ_{5}, χ_{3}χ_{4}ψ_{8} 120 Q(ζ_{5}, \sqrt{6})  2 1 2
(4^{∗}, 2) χ_{5}ψ_{8}, χ_{3}χ_{4} 120 Q(\sqrt{−(5 + \sqrt{5})}, \sqrt{3})  4 1 4
8  χ_{5}ψ_{8}, χ_{3}χ_{4}ψ_{8} 120 Q(\sqrt{−(5 + \sqrt{5})}, \sqrt{6})  4 1 4
χ_{5}, χ_{4}χ_{7}^{3}  140 Q(ζ_{5}, \sqrt{7})  2 1 2
χ_{3}ψ_{16}, χ_{3}χ_{4}χ_{5}^{2} 240 Q(\sqrt{−3(2 + \sqrt{2})}, \sqrt{15})  2 2 4
χ_{5}ψ_{8}, χ_{4}ψ_{8}χ_{7}^{3} 280 Q(\sqrt{−(5 + \sqrt{5})}, \sqrt{14})  4 1 4
χ_{5}ψ_{8}, χ_{4}  40 Q(\sqrt{−(5 + \sqrt{5})}, \sqrt{−1})  2 1 2
χ_{5}ψ_{8}, χ_{4}ψ_{8}  40 Q(\sqrt{−(5 + \sqrt{5})}, \sqrt{−2})  2 1 2
χ_{3}ψ_{16}, χ_{4}  48 Q(\sqrt{−3(2 + \sqrt{2})}, \sqrt{−1})  2 1 2
χ_{5}ψ_{16}, χ_{4}  80 Q(\sqrt{−(10 + 3\sqrt{10})}, \sqrt{−1})  2 2 4
χ_{5}ψ_{16}, χ_{4}ψ_{8}  80 Q(\sqrt{−(10 + 3\sqrt{10})}, \sqrt{−2})  2 2 4
χ_{4}ψ_{16}, χ_{4}χ_{5}^{2}  80 Q(i sin(π/8), \sqrt{−5})  1 2 2
(4^{∗}, 2^{∗}) ψ_{8}χ_{13}^{3}, χ_{4}  104 Q(\sqrt{−(13 + 3\sqrt{13})}, \sqrt{−1})  2 1 2
χ_{3}χ_{4}χ_{5}, χ_{4}ψ_{8} 120 Q(\sqrt{−3(5 + \sqrt{5})}, \sqrt{−2})  2 2 4
χ_{3}ψ_{16}, χ_{4}χ_{5}^{2} 240 Q(\sqrt{−3(2 + \sqrt{2})}, \sqrt{−5})  2 2 4
χ_{4}χ_{5}χ_{7}^{3}, χ_{4}ψ_{8} 280 Q(\sqrt{−7(5 + \sqrt{5})/2}, \sqrt{−2})  2 2 4
χ_{7}^{3}χ_{17}^{4}, χ_{3}  357 Q(\sqrt{−7(17 + 4\sqrt{17})}, \sqrt{−3}) 1 2 2
χ_{3}ψ_{16}, χ_{11}^{5} 528 Q(\sqrt{−3(2 + \sqrt{2})}, \sqrt{−11}) 1 2 2
χ_{7}^{3}χ_{17}^{4}, χ_{11}^{5} 1309 Q(\sqrt{−7(17 + 4\sqrt{17})}, \sqrt{−11}) 1 2 2

TABLE OF THE IMAGINARY ABELIAN FIELDS WITH g = h > 1 9

Continued

n Type  Generators  f  Simple expression  h^{−} h^{+} g
10 (2^{∗}, 5)  χ_{3}χ_{5}, ψ_{25}  75  2 1 2
(6^{∗}, 2)  χ_{7}, χ_{3}χ_{4}  84 Q(ζ_{7}, \sqrt{3})  2 1 2
(6^{∗}, 2^{∗})  χ_{7}, χ_{3}χ_{5}^{2}  105 Q(ζ_{7}, \sqrt{−15})  1 2 2
(4^{∗}, 3)  χ_{13}^{3}, χ_{7}^{2}χ_{13}^{4}  91  1 3 3
χ_{4}, χ_{3}ψ_{8}, ψ_{9}  72 Q(\sqrt{−1}, \sqrt{−6}, cos(2π/9)) 2 1 2
χ_{3}, χ_{4}χ_{7}^{3}, χ_{7}^{2}  84 Q(\sqrt{−3}, \sqrt{7}, cos(π/7))  2 1 2
(2^{∗}, 2, 3)  χ_{4}, χ_{3}χ_{7}^{3}, χ_{7}^{2}  84 Q(\sqrt{−1}, \sqrt{21}, cos(π/7))  2 1 2
χ_{7}^{3}, χ_{13}^{6}, χ_{7}^{4}χ_{13}^{4} 91  1 3 3
χ_{3}χ_{5}^{2}, χ_{3}χ_{7}^{3}, χ_{7}^{2} 105 Q(\sqrt{−1}, \sqrt{21}, cos(π/7))  2 1 2
χ_{3}, χ_{7}^{3}, χ_{7}^{4}ψ_{9}  63 Q(\sqrt{−3}, \sqrt{−7}, cos(2π/9)) 1 3 3
χ_{4}ψ_{8}, χ_{4}χ_{13}^{6}, χ_{13}^{4} 104  1 2 2
χ_{3}, χ_{5}^{2}χ_{7}^{3}, χ_{7}^{2} 105 Q(\sqrt{−3}, \sqrt{−35}, cos(π/7)) 1 2 2 12 χ_{4}, χ_{5}^{2}χ_{7}^{3}, χ_{7}^{2} 140 Q(\sqrt{−1}, \sqrt{−35}, cos(2π/7)) 1 2 2
χ_{4}ψ_{8}, χ_{3}χ_{4}χ_{7}^{3}, χ_{7}^{2} 168 Q(\sqrt{−2}, \sqrt{−21}, cos(2π/7)) 2 2 4
χ_{4}, χ_{3}χ_{5}^{2}, ψ_{9} 180 Q(\sqrt{−1}, \sqrt{−15}, cos(2π/9)) 1 2 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)) 1 2 2
χ_{7}^{3}, χ_{3}χ_{5}^{2}, ψ_{9} 315 Q(\sqrt{−7}, \sqrt{−15}, cos(2π/9)) 1 2 2
χ_{7}^{3}, χ_{4}χ_{13}^{6}, χ_{13}^{4} 364  1 2 2
χ_{4}, χ_{3}χ_{5}^{2}, χ_{7}^{2} 420 Q(\sqrt{−1}, \sqrt{−15}, cos(2π/7)) 1 2 2
χ_{4}ψ_{8}, χ_{3}χ_{5}^{2}, χ_{7}^{2} 840 Q(\sqrt{−2}, \sqrt{−15}, cos(2π/7)) 1 2 2
16 (8^{∗}, 2^{∗})  χ_{4}ψ_{32}, χ_{4}χ_{5}^{2} 160 Q(i sin(π/16), \sqrt{−5})  1 2 2