LIST OF KNOWN NONNORMAL

CM-FIELDS WITH CLASS NUMBER ONE

After proving the finiteness of imaginary abelian number fields with class numbers less than given bound, K. Uchida noted that the same argument shows that there exist only finitely many CM-fields with class numbers less than given bound of fixed degree. A. M. Odlyzko proved that the degrees of the CM-fields with class numbers less than given bound satusfying the condtion

the maximal totally real subfield is the top of a tower of fields F_{m} ⊃ ⋯ ⊃ F_{1} ⊃ F_{0} = Q such that each F_{i} is normal over the preceding one

are bounded, and that if we assume the truth of Generalized Riemann Hypothesis (GRH) or Artin’s conjecture, then we may drop the condition above. Recently V. K. Murty proved unconditionally the finiteness of the CM-fields with solvable normal closure and class numbers less than given bound. We list here known nonnormal CM-fields with class number one.

We use the following notations. Let K be a nonnormal CM-field with class number one and K_{+} its maximal totally real subfield. We denote by d_{K} and n the discriminant of K and the degree of K, respectively. When the Galois group of the normal closure of K is isomorphic to G, we say K to be of type G. We express K as K_{+}(\sqrt{−δ}), where δ is a totally positive algebraic integer of K_{+}. α_{i} denotes an algebraic numbers generating the ith totally real nonnormal cubic number field, where we consider that the totally real nonnormal cubic number fields are numbered by discriminants up to isomorphism. In the list of fields of type S_{4} × C_{2}, we take α_{i} as a root of a special cubic polynomial which is given below the list of fields.

Up to now (January 2003), the nonnormal CM-fields with class number one of degrees n with n ≦ 6 have been determined. Such fields of higher degrees have not been determined yet. The numbers of (isomorphism) such CM-fields of type G are as follows:

\begin{matrix}n & G & # \\ 4 & D_{4} & 37 \\ 6 & A_{4} × C_{2} & 19 \\ 6 & D_{6} & 124 \\ 6 & S_{4} × C_{2} & 233 \end{matrix}

As usual, C_{n} is the cyclic group of order n, D_{n} (n ≧ 3) is the dihedral group of order 2n, A_{4} (resp. S_{4}) is the alternating (resp. symmetric) group of degree four.

Typeset by AMS-TEX

(I) n = 4. S. Louboutin and R. Okazaki have determined the nonnormal quartic CM-fields

with class number one. There exist exactly 37 nonisomorphic such fields. All of them are of type D_{4}, that is, their normal closures have Galois group isomorphic to D_{4}, the dihedral group of order eight.

Type D_{4}. (37 fields (up to isomorphism))

K  K_{+}  d_{K}  rd_{K}
Q(\sqrt{−(13 + \sqrt{5})/2})  Q(\sqrt{5})  1025 = 5^{2} · 41  5.658
Q(\sqrt{−(5 + 2\sqrt{2})})  Q(\sqrt{2})  1088 = 2^{6} · 17  5.743
Q(\sqrt{−(17 + 3\sqrt{5})/2}) Q(\sqrt{5})  1525 = 5^{2} · 61  6.249
Q(\sqrt{−(5 + \sqrt{17})/2})  Q(\sqrt{17})  2312 = 2^{3} · 17^{2} 6.934
Q(\sqrt{−(21 + \sqrt{5})/2})  Q(\sqrt{5})  2725 = 5^{2} · 109 7.225
Q(\sqrt{−(9 + \sqrt{13})/2})  Q(\sqrt{13})  2873 = 13^{2} · 17 7.321
Q(\sqrt{−(13 + 2\sqrt{5})})  Q(\sqrt{5})  3725 = 5^{2} · 149 7.812
Q(\sqrt{−(9 + 2\sqrt{17})})  Q(\sqrt{17})  3757 = 17^{2} · 13 7.829
Q(\sqrt{−(9 + 2\sqrt{2})})  Q(\sqrt{2})  4672 = 2^{6} · 73  8.267
Q(\sqrt{−(9 + 2\sqrt{13})})  Q(\sqrt{13})  4901 = 13^{2} · 29 8.367
Q(\sqrt{−(11 + 4\sqrt{2})})  Q(\sqrt{2})  5696 = 2^{6} · 89  8.687
Q(\sqrt{−(17 + 2\sqrt{5})})  Q(\sqrt{5})  6725 = 5^{2} · 269 9.055
Q(\sqrt{−(13 + 2\sqrt{41})})  Q(\sqrt{41})  8405 = 5 · 41^{2}  9.574
Q(\sqrt{−(41 + 5\sqrt{5})/2}) Q(\sqrt{5})  9725 = 5^{2} · 389 9.930
Q(\sqrt{−(9 + \sqrt{29})/2})  Q(\sqrt{29}) 10933 = 13 · 29^{2} 10.225
Q(\sqrt{−(19 + 8\sqrt{2})})  Q(\sqrt{2})  14912 = 2^{6} · 233 11.050
Q(\sqrt{−(17 + 2\sqrt{2})})  Q(\sqrt{2})  17984 = 2^{6} · 281 11.580
Q(\sqrt{−(17 + 2\sqrt{61})})  Q(\sqrt{61}) 18605 = 5 · 61^{2} 11.679
Q(\sqrt{−(25 + 6\sqrt{13})})  Q(\sqrt{13}) 26533 = 13^{2} · 157 12.762
Q(\sqrt{−(29 + 3\sqrt{13})/2}) Q(\sqrt{13}) 30589 = 13^{2} · 181 13.224
Q(\sqrt{−(35 + 8\sqrt{17})})  Q(\sqrt{17}) 39593 = 17^{2} · 137 14.106
Q(\sqrt{−(9 + \sqrt{73})/2})  Q(\sqrt{73}) 42632 = 2^{3} · 73^{2} 14.369
Q(\sqrt{−(13 + 2\sqrt{29})})  Q(\sqrt{29}) 44573 = 29^{2} · 53 14.530
Q(\sqrt{−(21 + 2\sqrt{109})}) Q(\sqrt{109}) 59405 = 5 · 109^{2} 15.611
Q(\sqrt{−(11 + \sqrt{89})/2}) Q(\sqrt{89}) 63368 = 2^{3} · 89^{2} 15.866
Q(\sqrt{−(23 + 4\sqrt{17})})  Q(\sqrt{17}) 74273 = 17^{2} · 257 16.508
Q(\sqrt{−(13 + \sqrt{53})/2}) Q(\sqrt{53}) 81461 = 29 · 53^{2} 16.894

Continued

K  K_{+}  d_{K}  rd_{K}
Q(\sqrt{−(13 + \sqrt{149})/2}) Q(\sqrt{149}) 111005 = 5 · 149^{2} 18.253
Q(\sqrt{−(35 + \sqrt{137})/2}) Q(\sqrt{137}) 319073 = 17 · 137^{2} 23.778
Q(\sqrt{−(25 + \sqrt{157})/2}) Q(\sqrt{157}) 320437 = 13 · 157^{2} 23.792
Q(\sqrt{−(17 + \sqrt{269})/2}) Q(\sqrt{269}) 361805 = 5 · 269^{2} 24.525
Q(\sqrt{−(29 + 2\sqrt{181})}) Q(\sqrt{181}) 425893 = 13 · 181^{2} 25.546
Q(\sqrt{−(19 + \sqrt{233})/2}) Q(\sqrt{233}) 434312 = 2^{3} · 233^{2} 25.671
Q(\sqrt{−(103 + 12\sqrt{73})}) Q(\sqrt{73}) 516913 = 73^{2} · 97 26.813
Q(\sqrt{−(17 + \sqrt{281})/2}) Q(\sqrt{281}) 631688 = 2^{3} · 281^{2} 28.191
Q(\sqrt{−(103 + \sqrt{97})/2}) Q(\sqrt{97}) 686857 = 73 · 97^{2} 28.788
Q(\sqrt{−(41 + 2\sqrt{389})}) Q(\sqrt{389}) 756605 = 5 · 389^{2} 29.492

There exists exactly one other nonnormal quartic CM-field with relative class number one.

K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−(23 + 2\sqrt{17})/2}) Q(\sqrt{257}) 1122833 = 17 · 257^{2} 32.552

For this field, the class number of K_{+} is three.

These 38 fields with relative class number one are divided into 19 pairs of fields whose normal closures coincide; Any nonnormal quartic CM-field K with odd relative class number is of the form Q(\sqrt{p}, \sqrt{−ξ}), where p is a prime number ≢ 3 (mod 4) and ξ is a totally positive generator of a prime ideal above a prime number q that splits in Q(\sqrt{p}) Then the normal closure N of K is a dihedral octic CM-field that is a cyclic quartic extension of the real quadratic number field Q(\sqrt{pq}) unramified at all finite primes, and any one K_{∗} of the two nonnormal quartic subfields of N not isomorphic to K is of the form Q(\sqrt{q}, \sqrt{−η}), where η is a totally positive generator of a prime ideal above a prime number p, that splits in Q(\sqrt{q}). Moreover, the relations h^{−}(N) = h^{−}(K)^{2} = h^{−}(K_{∗})^{2} hold. Therefore these 19 pairs correspond one to one to the dihedral octic CM-fields with relative class number one, and their simple expressions as pairs {p, q} are as follows:

{2, 17}, {2, 73}, {2, 89}, {2, 233}, {2, 281}, {5, 41}, {5, 61}, {5, 109}, {5, 149}, {5, 269}, {5, 389}, {13, 17}, {13, 29}, {13, 157}, {13, 181}, {17, 137}, {17, 257}, {29, 53}, {73, 97}.

(II) n = 6.

G. Boutteaux and S. R. Louboutin announced that G. Boutteaux has completed the determination of the nonnormal sextic CM-fields with class number one under supervision by S. R. Louboutin. There exist exactly 386 nonisomorphic nonnormal sextic CM-fields with class number one; 19 of them are of type A_{4} × C_{2}, 134 of them are of type D_{6}, and the other 233 are of type S_{4} × C_{2}.

(i) The case where K_{+} is normal (cyclic cubic).

The nonnormal sextic CM-fields whose maximal totally real subfields are normal are of type A_{4} × C_{2}. There exist exactly 19 nonisomorphic nonnormal sextic CMfields of type A_{4} × C_{2} with class number one. Any nonnormal sextic CM-field K of type A_{4} × C_{2} with odd relative class number is of the form K_{+}(\sqrt{−δ}), where K_{+} is a cyclic cubic number field with odd narrow class number h_{+}(K_{+}) and α is a totally positive generator of the h_{+}(K_{+})th power of a prime ideal 𝔔 in K_{+} of degree one. (They call such a field a simplest sextic CM-field.) The isomorphism class of such a field is determied by the conductor f_{K_{+}} of K_{+} and the rational prime q below 𝔔. For each such K with class number one, in addition to d_{K} and rd_{K}, we give f_{K_{+}}, q and the minimal polynomial P (X) of some α.

Type A_{4} × C_{2}. (19 fields (up to isomorphism))

f_{K_{+}}  q  P (X)  d_{K}  rd_{K}
7 167 X^{3} − 18X^{2} + 101X − 167  −400967 = −7^{4} · 167 8.587
9 71 X^{3} − 30X^{2} + 117X − 71 −465831 = −3^{8} · 71 8.804
7 239 X^{3} − 19X^{2} + 118X − 239  −573839 = −7^{4} · 239 9.115
7 251 X^{3} − 22X^{2} + 145X − 251  −602651 = −7^{4} · 251 9.190
7 379 X^{3} − 26X^{2} + 181X − 379  −909979 = −7^{4} · 379 9.844
7 491 X^{3} − 26X^{2} + 209X − 491  −1178891 = −7^{4} · 491 10.278
9 199 X^{3} − 39X^{2} + 318X − 199  −1305639 = −3^{8} · 199 10.454
7 547 X^{3} − 27X^{2} + 222X − 547  −1313347 = −7^{4} · 547 10.464
13  47 X^{3} − 15X^{2} + 62X − 47  −1342367 = −13^{4} · 47 10.502
13  79 X^{3} − 14X^{2} + 61X − 79  −2256319 = −13^{4} · 79 11.452
9 379 X^{3} − 30X^{2} + 237X − 379  −2486619 = −3^{8} · 379 11.639
7 1051 X^{3} − 34X^{2} + 341X − 1051 −2523451 = −7^{4} · 1051 11.668
9 523 X^{3} − 57X^{2} + 507X − 523  −3431403 = −3^{8} · 523 12.281
19 31 X^{3} − 11X^{2} + 34X − 31 −4039951 = −19^{4} · 31 12.620
9 739 X^{3} − 33X^{2} + 315X − 739  −4848579 = −3^{8} · 739 13.009
31  31 X^{3} − 12X^{2} + 17X − 2  −7388168 = −2^{3} · 31^{4} 13.955
19  83 X^{3} − 18X^{2} + 89X − 83  −10816643 = −19^{4} · 83 14.871
37  11 X^{3} − 10X^{2} + 21X − 11  −20615771 = −11 · 37^{4} 16.558
61  3 X^{3} − 15X^{2} + 14X − 3  −41537523 = −3 · 61^{4} 18.609

(ii) The case where K_{+} is nonnormal.

Nonnormal sextic CM-fields whose maximal totally real subfields are nonnormal are of type D_{6} or S_{4} ×C_{2} according as they contain an imaginary quadratic number field or do not.

There exist exactly 134 nonisomorphic nonnormal sextic CM-fields of type D_{6} with class number one. Each nonnormal sextic CM-field K of type D_{6} is the compositum of a nonnormal totally real cubic number field K_{+} with an imaginary quadratic number field F and if K has class number one, then also F has class number one. For K with class number one, the isomorphism class of K_{+} is uniquely determined by its discriminant. Moreover, in this case only one rational finite prime q is ramified in F. 134 fields are divided into three by the factorization type (abbrev. f-type) of q in K_{+}, 3 (inert), 1^{3} (totally ramified), or 1^{2}1 (partially ramified). (111 can never occur (q can never split completely in K_{+}).)

Type D_{6} and f-type 3 (5 fields (up to isomorphism))

K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−3}) Q(α_{1})  −591408 = −2^{4} · 3^{3} · 37^{2}  9.161
K_{+}(\sqrt{−3}) Q(α_{4}) −2696112 = −2^{4} · 3^{3} · 79^{2}  11.797
K_{+}(\sqrt{−1}) Q(α_{5}) −6594624 = −2^{6} · 3^{2} · 107^{2} 13.694
K_{+}(\sqrt{−3}) Q(α_{10}) −8710848 = −2^{6} · 3^{3} · 71^{2}  14.344
K_{+}(\sqrt{−3}) Q(α_{20}) −23857200 = −2^{4} · 3^{3} · 5^{2} · 47^{2} 16.966

Type D_{6} and f-type 1^{3} (26 fields (up to isomorphism))

K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−1}) Q(α_{1})  −350464 = −2^{8} · 37^{2}  8.396
K_{+}(\sqrt{−3}) Q(α_{11}) −1156923 = −3^{7} · 23^{2}  10.245
K_{+}(\sqrt{−3}) Q(α_{14}) −1714608 = −2^{4} · 3^{7} · 7^{2} 10.940
K_{+}(\sqrt{−3}) Q(α_{18}) −2101707 = −3^{7} · 31^{2}  11.317
K_{+}(\sqrt{−1}) Q(α_{4}) −2611456 = −2^{8} · 101^{2} 11.734
K_{+}(\sqrt{−2}) Q(α_{1}) −2803712 = −2^{11} · 37^{2}  11.874
K_{+}(\sqrt{−1}) Q(α_{9}) −5089536 = −2^{8} · 3^{2} · 47^{2} 13.115
K_{+}(\sqrt{−3}) Q(α_{44}) −7873200 = −2^{4} · 3^{9} · 5^{2} 14.104
K_{+}(\sqrt{−1}) Q(α_{14}) −9144576 = −2^{8} · 3^{6} · 7^{2} 14.460
K_{+}(\sqrt{−3}) Q(α_{53}) −11337408 = −2^{6} · 3^{11}  14.988
K_{+}(\sqrt{−3}) Q(α_{64}) −15066243 = −3^{7} · 83^{2}  15.715
K_{+}(\sqrt{−3}) Q(α_{77}) −21870000 = −2^{4} · 3^{7} · 5^{4} 16.722
K_{+}(\sqrt{−3}) Q(α_{82}) −23654592 = −2^{6} · 3^{7} · 13^{2} 16.942
K_{+}(\sqrt{−11}) Q(α_{42}) −27217619 = −11^{5} · 13^{2} 17.343
K_{+}(\sqrt{−3}) Q(α_{93}) −29428272 = −2^{4} · 3^{7} · 29^{2} 17.570

6 LIST OF KNOWN NONNORMAL CM-FIELDS WITH CLASS NUMBER ONE

Continued

K K_{+} d_{K} rd_{K}
K_{+}(\sqrt{−1}) Q(α_{40})  −37161216 = −2^{8} · 3^{2} · 127^{2} 18.267
K_{+}(\sqrt{−3}) Q(α_{131})  −50528448 = −2^{6} · 3^{7} · 19^{2}  19.227
K_{+}(\sqrt{−3}) Q(α_{160})  −70848800 = −2^{4} · 3^{11} · 5^{2}  20.342
K_{+}(\sqrt{−3}) Q(α_{202}) −105850800 = −2^{4} · 3^{7} · 5^{5} · 11^{2} 21.749
K_{+}(\sqrt{−3}) Q(α_{226}) −130205232 = −2^{4} · 3^{7} · 611^{2} 22.513
K_{+}(\sqrt{−1}) Q(α_{92}) −156150016 = −2^{8} · 11^{2} · 71^{2} 23.205
K_{+}(\sqrt{−3}) Q(α_{298}) −206122563 = −3^{7} · 307^{2}  24.304
K_{+}(\sqrt{−3}) Q(α_{335}) −251332227 = −3^{9} · 113^{2}  25.121
K_{+}(\sqrt{−3}) Q(α_{457}) −423403200 = −2^{6} · 3^{7} · 5^{2} · 11^{2} 27.402
K_{+}(\sqrt{−3}) Q(α_{491}) −487228608 = −2^{6} · 3^{7} · 59^{2}  28.051
K_{+}(\sqrt{−3}) Q(α_{837}) −1246644675 = −3^{7} · 5^{2} · 151^{2} 32.806

Type D_{6} and f-type 1^{2}1 (103 fields (up to isomorphism))

K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−3}) Q(α_{5})  −309123 = −3^{3} · 107^{2}  8.222
K_{+}(\sqrt{−1}) Q(α_{4})  −399424 = −2^{6} · 79^{2}  8.581
K_{+}(\sqrt{−3}) Q(α_{9})  −954288 = −2^{4} · 3^{3} · 47^{2} 9.922
K_{+}(\sqrt{−7}) Q(α_{7}) −1539727 = −7^{3} · 67^{2}  10.745
K_{+}(\sqrt{−11}) Q(α_{8}) −2461019 = −11^{3} · 43^{2}  11.619
K_{+}(\sqrt{−2}) Q(α_{10}) −2580992 = −2^{9} · 71^{2}  11.712
K_{+}(\sqrt{−3}) Q(α_{22}) −2958147 = −3^{3} · 331^{2}  11.981
K_{+}(\sqrt{−1}) Q(α_{20}) −3534400 = −2^{6} · 5^{2} · 47^{2} 12.342
K_{+}(\sqrt{−3}) Q(α_{25}) −3636603 = −3^{3} · 367^{2}  12.400
K_{+}(\sqrt{−7}) Q(α_{14}) −4000752 = −2^{4} · 3^{6} · 7^{3} 12.599
K_{+}(\sqrt{−3}) Q(α_{35}) −6091875 = −3^{3} · 5^{4} · 19^{2} 13.514
K_{+}(\sqrt{−3}) Q(α_{40}) −6967728 = −2^{4} · 3^{3} · 127^{2} 13.820
K_{+}(\sqrt{−43}) Q(α_{8}) −9620347 = −11^{2} · 43^{3}  14.583
K_{+}(\sqrt{−1}) Q(α_{45}) −11669056 = −2^{6} · 7^{2} · 61^{2} 15.060
K_{+}(\sqrt{−1}) Q(α_{47}) −12559936 = −2^{6} · 443^{2}  15.246
K_{+}(\sqrt{−2}) Q(α_{30}) −13603328 = −2^{9} · 163^{2}  15.450
K_{+}(\sqrt{−67}) Q(α_{7}) −14737387 = −7^{2} · 67^{3}  15.658
K_{+}(\sqrt{−3}) Q(α_{70}) −18825075 = −3^{3} · 5^{2} · 167^{2} 16.310
K_{+}(\sqrt{−1}) Q(α_{74}) −27793984 = −2^{6} · 659^{2}  17.404
K_{+}(\sqrt{−1}) Q(α_{77}) −29160000 = −2^{6} · 3^{6} · 5^{4} 17.544

LIST OF KNOWN NONNORMAL CM-FIELDS WITH CLASS NUMBER ONE 7

Continued.

K K_{+} d_{K} rd_{K}
K_{+}(\sqrt{−3}) Q(α_{95}) −29654208 = −2^{6} · 3^{3} · 131^{2}  17.593
K_{+}(\sqrt{−2}) Q(α_{53}) −30233088 = −2^{9} · 3^{10}  17.650
K_{+}(\sqrt{−3}) Q(α_{99}) −31726512 = −2^{11} · 3^{3} · 271^{2}  17.792
K_{+}(\sqrt{−7}) Q(α_{60}) −33175303 = −7^{3} · 311^{2}  17.925
K_{+}(\sqrt{−7}) Q(α_{63}) −34904023 = −7^{3} · 11^{2} · 29^{2}  18.077
K_{+}(\sqrt{−3}) Q(α_{109}) −37594800 = −2^{4} · 3^{3} · 5^{2} · 59^{2} 18.302
K_{+}(\sqrt{−3}) Q(α_{111}) −38363328 = −2^{6} · 3^{3} · 149^{2}  18.364
K_{+}(\sqrt{−19}) Q(α_{35}) −38581875 = −3^{2} · 5^{4} · 19^{3}  18.382
K_{+}(\sqrt{−3}) Q(α_{120}) −45000387 = −3^{3} · 1291^{2}  18.859
K_{+}(\sqrt{−11}) Q(α_{56}) −45062336 = −2^{6} · 11^{3} · 23^{2}  18.864
K_{+}(\sqrt{−11}) Q(α_{59}) −48556211 = −11^{3} · 191^{2}  19.100
K_{+}(\sqrt{−1}) Q(α_{114}) −51724864 = −2^{6} · 29^{2} · 31^{2}  19.302
K_{+}(\sqrt{−19}) Q(α_{63}) −54849179 = −7^{2} · 11^{3} · 29^{2}  19.492
K_{+}(\sqrt{−3}) Q(α_{149}) −66044592 = −2^{4} · 3^{3} · 17^{2} · 23^{2} 20.105
K_{+}(\sqrt{−3}) Q(α_{152}) −68087088 = −2^{4} · 3^{3} · 397^{2}  20.207
K_{+}(\sqrt{−1}) Q(α_{139}) −76177984 = −2^{6} · 1091^{2}  20.589
K_{+}(\sqrt{−1}) Q(α_{156}) −93857344 = −2^{6} · 7^{2} · 173^{2}  21.317
K_{+}(\sqrt{−2}) Q(α_{118}) −111661568 = −2^{9} · 467^{2}  21.944
K_{+}(\sqrt{−1}) Q(α_{175}) −114746944 = −2^{6} · 13^{2} · 103^{2} 22.043
K_{+}(\sqrt{−7}) Q(α_{132}) −123068743 = −7^{3} · 599^{2}  22.302
K_{+}(\sqrt{−3}) Q(α_{218}) −123649200 = −2^{4} · 3^{3} · 5^{2} · 107^{2} 22.320
K_{+}(\sqrt{−3}) Q(α_{242}) −148347072 = −2^{6} · 3^{3} · 293^{2}  23.008
K_{+}(\sqrt{−7}) Q(α_{146}) −150772167 = −3^{2} · 7^{3} · 13^{2} · 17^{2} 23.070
K_{+}(\sqrt{−3}) Q(α_{259}) −167133888 = −2^{6} · 3^{3} · 311^{2}  23.469
K_{+}(\sqrt{−19}) Q(α_{90}) −173402379 = −3^{2} · 19^{3} · 53^{2}  23.614
K_{+}(\sqrt{−7}) Q(α_{165}) −181285447 = −7^{3} · 727^{2}  23.789
K_{+}(\sqrt{−1}) Q(α_{245}) −200732224 = −2^{6} · 7^{2} · 11^{2} · 23^{2} 24.197
K_{+}(\sqrt{−7}) Q(α_{177}) −201783127 = −3^{2} · 7^{3} · 13^{2} · 59^{2} 24.218
K_{+}(\sqrt{−3}) Q(α_{294}) −202705200 = −2^{4} · 3^{3} · 5^{2} · 137^{2} 24.236
K_{+}(\sqrt{−1}) Q(α_{250}) −209902144 = −2^{6} · 1811^{2}  24.378
K_{+}(\sqrt{−3}) Q(α_{302}) −215324352 = −2^{6} · 3^{3} · 353^{2}  24.482
K_{+}(\sqrt{−1}) Q(α_{254}) −219276864 = −2^{6} · 3^{2} · 617^{2}  24.556
K_{+}(\sqrt{−19}) Q(α_{107}) −232218304 = −2^{6} · 19^{3} · 23^{2}  24.792
K_{+}(\sqrt{−3}) Q(α_{336}) −253478592 = −2^{6} · 3^{3} · 383^{2}  25.156
K_{+}(\sqrt{−7}) Q(α_{209}) −150772167 = −3^{2} · 7^{3} · 293^{2}  25.344

8 LIST OF KNOWN NONNORMAL CM-FIELDS WITH CLASS NUMBER ONE

Continued.

K K_{+} d_{K} rd_{K}
K_{+}(\sqrt{−163}) Q(α_{30}) −277167808 = −2^{6} · 163^{3}  25.534
K_{+}(\sqrt{−1})  Q(α_{308}) −292820544 = −2^{6} · 3^{2} · 23^{2} · 31^{2} 25.769
K_{+}(\sqrt{−7}) Q(α_{227}) −305012407 = −7^{3} · 23^{2} · 41^{2} 25.944
K_{+}(\sqrt{−3})  Q(α_{373}) −312120000 = −2^{6} · 3^{3} · 5^{4} · 17^{2} 26.044
K_{+}(\sqrt{−11}) Q(α_{176}) −316969664 = −2^{6} · 11^{3} · 61^{2}  26.111
K_{+}(\sqrt{−19}) Q(α_{131}) −320013504 = −2^{6} · 3^{6} · 19^{3}  26.153
K_{+}(\sqrt{−3}) Q(α_{392}) −339692643 = −3^{3} · 3547^{2} 26.414
K_{+}(\sqrt{−3})  Q(α_{399}) −350698032 = −2^{4} · 3^{3} · 17^{2} · 53^{2} 26.555
K_{+}(\sqrt{−1}) Q(α_{352}) −374499904 = −2^{6} · 41^{2} · 59^{2} 26.847
K_{+}(\sqrt{−7})  Q(α_{272}) −411265575 = −3^{2} · 5^{2} · 7^{3} · 73^{2} 27.270
K_{+}(\sqrt{−3})  Q(α_{469}) −447691968 = −2^{6} · 3^{3} · 509^{2}  27.658
K_{+}(\sqrt{−7})  Q(α_{292}) −460745383 = −7^{3} · 19^{2} · 61^{2}  27.791
K_{+}(\sqrt{−7})  Q(α_{305}) −506345175 = −3^{10} · 5^{2} · 7^{3}  28.231
K_{+}(\sqrt{−11}) Q(α_{245}) −552013616 = −2^{4} · 7^{2} · 11^{3} · 23^{2} 28.641
K_{+}(\sqrt{−1}) Q(α_{458}) −564917824 = −2^{6} · 2971^{2} 28.751
K_{+}(\sqrt{−19}) Q(α_{183}) −580826979 = −3^{2} · 19^{3} · 97^{2}  28.885
K_{+}(\sqrt{−19}) Q(α_{188}) −600958144 = −2^{6} · 19^{3} · 37^{2}  29.049
K_{+}(\sqrt{−3})  Q(α_{566}) −632491200 = −2^{6} · 3^{3} · 5^{2} · 11^{4} 29.298
K_{+}(\sqrt{−2})  Q(α_{327}) −636531200 = −2^{9} · 5^{2} · 223^{2}  29.329
K_{+}(\sqrt{−7})  Q(α_{363}) −686763175 = −5^{2} · 7^{3} · 283^{2}  29.703
K_{+}(\sqrt{−7})  Q(α_{384}) −750292263 = −3^{2} · 7^{3} · 17^{2} · 29^{2} 30.144
K_{+}(\sqrt{−1})  Q(α_{553}) −819791424 = −2^{6} · 3^{2} · 1193^{2} 30.592
K_{+}(\sqrt{−3}) Q(α_{669}) −835568163 = −3^{3} · 5563^{2} 30.690
K_{+}(\sqrt{−3})  Q(α_{675}) −852778800 = −2^{4} · 3^{3} · 5^{2} · 281^{2} 30.794
K_{+}(\sqrt{−1})  Q(α_{579})  −89472744 = −2^{6} · 3739^{2}  31.041
K_{+}(\sqrt{−2})  Q(α_{412}) −973640192 = −2^{9} · 7^{2} · 197^{2}  31.482
K_{+}(\sqrt{−3})  Q(α_{729}) −993829203 = −3^{3} · 6067^{2}  31.590
K_{+}(\sqrt{−1})  Q(α_{631}) −1029639744 = −2^{6} · 3^{2} · 7^{2} · 191^{2} 31.777
K_{+}(\sqrt{−2}) Q(α_{439}) −1077965312 = −2^{9} · 1451^{2} 32.020
K_{+}(\sqrt{−3})  Q(α_{760}) −1085281200 = −2^{4} · 3^{3} · 5^{2} · 317^{2} 32.057
K_{+}(\sqrt{−3})  Q(α_{812}) −1208775987 = −3^{3} · 6691^{2}  32.638
K_{+}(\sqrt{−1})  Q(α_{743}) −1384435264 = −2^{6} · 4651^{2}  33.384
K_{+}(\sqrt{−7})  Q(α_{556}) −1448497575 = −3^{2} · 5^{2} · 7^{3} · 137^{2} 33.637
K_{+}(\sqrt{−2})  Q(α_{552}) −1635004928 = −2^{9} · 1787^{2}  34.323

LIST OF KNOWN NONNORMAL CM-FIELDS WITH CLASS NUMBER ONE 9

Continued.

K K_{+} d_{K} rd_{K}
K_{+}(\sqrt{−2}) Q(α_{572}) −1724019200 = −2^{9} · 5^{2} · 367^{2}  34.627
K_{+}(\sqrt{−1}) Q(α_{879}) −1818851904 = −2^{6} · 3^{2} · 1777^{2}  34.938
K_{+}(\sqrt{−7}) Q(α_{671}) −1960890183 = −3^{2} · 7^{3} · 797^{2}  35.378
K_{+}(\sqrt{−43}) Q(α_{249}) −2244005568 = −2^{6} · 3^{2} · 7^{2} · 43^{2} 36.182
K_{+}(\sqrt{−11}) Q(α_{629}) −2802288731 = −11^{3} · 1451^{2}  37.547
K_{+}(\sqrt{−7}) Q(α_{848}) −2954679175 = −5^{2} · 7^{3} · 587^{2}  37.880
K_{+}(\sqrt{−11}) Q(α_{685}) −3222532016 = −2^{4} · 11^{3} · 389^{2}  38.432
K_{+}(\sqrt{−7}) Q(α_{1048}) −4462584007 = −7^{3} · 3607^{2}  40.575
K_{+}(\sqrt{−43}) Q(α_{421}) −5333409067 = −7^{2} · 37^{2} · 43^{2}  41.799
K_{+}(\sqrt{−2}) Q(α_{1205}) −6499824128 = −2^{9} · 7^{2} · 509^{2}  43.199
K_{+}(\sqrt{−7}) Q(α_{1303}) −6661625607 = −3^{2} · 7^{3} · 13^{2} · 113^{2} 43.377
K_{+}(\sqrt{−11}) Q(α_{1125}) −8048132411 = − · 11^{3} · 2459^{2}  44.766
K_{+}(\sqrt{−11}) Q(α_{1633}) −15558805691 = −11^{3} · 13^{2} · 263^{2} 49.964
K_{+}(\sqrt{−19}) Q(α_{1705}) −28993925824 = −2^{6} · 19^{3} · 257^{2}  55.426