Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Unit group torsion |
Unit group rank |
Regulator |
13.13.50359924122392641.1 |
$x^{13} - 2 x^{12} - 11 x^{11} + 23 x^{10} + 39 x^{9} - 91 x^{8} - 44 x^{7} + 145 x^{6} - 6 x^{5} - 89 x^{4} + 29 x^{3} + 14 x^{2} - 8 x + 1$ |
$13$ |
[13,0] |
$50359924122392641$ |
$1$ |
$19.2652999174$ |
$224410169.38274574$ |
|
|
✓ |
$S_{13}$ (as 13T9) |
trivial |
$2$ |
$12$ |
$8144.93556547$ |
13.13.74057741281094693.1 |
$x^{13} - 3 x^{12} - 8 x^{11} + 27 x^{10} + 22 x^{9} - 89 x^{8} - 23 x^{7} + 132 x^{6} + 5 x^{5} - 87 x^{4} + 4 x^{3} + 21 x^{2} - 2 x - 1$ |
$13$ |
[13,0] |
$74057741281094693$ |
$1$ |
$19.8453730138$ |
$272135520.06508577$ |
|
|
✓ |
$S_{13}$ (as 13T9) |
trivial |
$2$ |
$12$ |
$10098.9038604$ |
13.13.491...641.1 |
$x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} - 215 x^{4} - 291 x^{3} + 68 x^{2} + 10 x - 1$ |
$13$ |
[13,0] |
$53^{12}$ |
$1$ |
$39.0516388455$ |
$39.05163884547241$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$1314145.36669$ |
13.13.590...441.1 |
$x^{13} - x^{12} - 36 x^{11} + 77 x^{10} + 365 x^{9} - 1193 x^{8} - 617 x^{7} + 5541 x^{6} - 4414 x^{5} - 4575 x^{4} + 6321 x^{3} + 411 x^{2} - 2196 x + 293$ |
$13$ |
[13,0] |
$79^{12}$ |
$1$ |
$56.4489381067$ |
$56.44893810672074$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$14045368.3938$ |
13.13.145...912.1 |
$x^{13} - 26 x^{11} + 260 x^{9} - 1248 x^{7} + 2912 x^{5} - 2912 x^{3} + 832 x - 176$ |
$13$ |
[13,0] |
$2^{12}\cdot 7^{6}\cdot 13^{13}$ |
$3$ |
$60.5149172415$ |
$78.14721442881216$ |
|
|
? |
$F_{13}$ (as 13T6) |
trivial |
$2$ |
$12$ |
$87393029.0991$ |
13.13.282...601.1 |
$x^{13} - 3 x^{12} - 41 x^{11} + 155 x^{10} + 376 x^{9} - 2028 x^{8} - 13 x^{7} + 8927 x^{6} - 7711 x^{5} - 13367 x^{4} + 20364 x^{3} + 1404 x^{2} - 13408 x + 5312$ |
$13$ |
[13,0] |
$8101^{6}$ |
$1$ |
$63.6708880391$ |
$90.00555538409837$ |
|
|
|
$D_{13}$ (as 13T2) |
trivial |
$2$ |
$12$ |
$235421273.11$ |
13.13.353...881.1 |
$x^{13} - 39 x^{11} + 507 x^{9} - 156 x^{8} - 2925 x^{7} + 1872 x^{6} + 7605 x^{5} - 7488 x^{4} - 6435 x^{3} + 10062 x^{2} - 2691 x - 306$ |
$13$ |
[13,0] |
$3^{12}\cdot 13^{16}$ |
$2$ |
$64.7778937113$ |
$73.88429313471292$ |
|
|
|
$C_{13}:C_3$ (as 13T3) |
trivial |
$2$ |
$12$ |
$166789217.838$ |
13.13.361...641.1 |
$x^{13} - x^{12} - 50 x^{11} + 25 x^{10} + 722 x^{9} - 226 x^{8} - 4207 x^{7} + 1158 x^{6} + 10465 x^{5} - 2535 x^{4} - 9399 x^{3} + 1079 x^{2} + 1316 x - 49$ |
$13$ |
[13,0] |
$23^{6}\cdot 367^{6}$ |
$2$ |
$64.8906015414$ |
$91.87491496594704$ |
|
|
|
$D_{13}$ (as 13T2) |
trivial |
$2$ |
$12$ |
$100827679.942$ |
13.13.255...961.1 |
$x^{13} - x^{12} - 60 x^{11} + 27 x^{10} + 1199 x^{9} - 33 x^{8} - 9610 x^{7} - 3352 x^{6} + 33548 x^{5} + 20328 x^{4} - 47723 x^{3} - 34869 x^{2} + 21271 x + 15667$ |
$13$ |
[13,0] |
$131^{12}$ |
$1$ |
$90.0335289186$ |
$90.03352891861981$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$292369424.433$ |
13.13.224...601.1 |
$x^{13} - x^{12} - 72 x^{11} + 129 x^{10} + 1672 x^{9} - 3386 x^{8} - 16810 x^{7} + 32367 x^{6} + 81708 x^{5} - 121902 x^{4} - 196272 x^{3} + 127412 x^{2} + 217458 x + 61399$ |
$13$ |
[13,0] |
$157^{12}$ |
$1$ |
$106.410453613$ |
$106.41045361273743$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$1334709178.87$ |
13.13.302...000.1 |
$x^{13} - 65 x^{11} + 1625 x^{9} - 19500 x^{7} + 113750 x^{5} - 284375 x^{3} + 203125 x - 69500$ |
$13$ |
[13,0] |
$2^{12}\cdot 5^{12}\cdot 13^{13}$ |
$3$ |
$108.898093209$ |
$137.63336134966406$ |
|
|
? |
$F_{13}$ (as 13T6) |
trivial |
$2$ |
$12$ |
$4111386187.29$ |
13.13.542...361.1 |
$x^{13} - 78 x^{11} - 65 x^{10} + 2080 x^{9} + 2457 x^{8} - 24128 x^{7} - 27027 x^{6} + 137683 x^{5} + 110214 x^{4} - 376064 x^{3} - 128206 x^{2} + 363883 x - 12167$ |
$13$ |
[13,0] |
$13^{24}$ |
$1$ |
$113.896611641$ |
$113.89661164093096$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$2733056590.62$ |
13.13.910...125.1 |
$x^{13} - 130 x^{11} - 390 x^{10} + 3900 x^{9} + 21255 x^{8} + 10985 x^{7} - 119145 x^{6} - 213785 x^{5} + 106470 x^{4} + 499395 x^{3} + 197730 x^{2} - 296595 x - 205335$ |
$13$ |
[13,0] |
$3^{6}\cdot 5^{12}\cdot 13^{15}$ |
$3$ |
$141.49208407$ |
$171.13897383055289$ |
|
|
|
$C_{13}:C_4$ (as 13T4) |
trivial |
$2$ |
$12$ |
$254843353531$ |
13.13.567...184.1 |
$x^{13} - 117 x^{11} - 26 x^{10} + 3692 x^{9} + 2938 x^{8} - 38987 x^{7} - 54314 x^{6} + 137670 x^{5} + 290524 x^{4} - 5824 x^{3} - 298948 x^{2} - 169403 x - 13406$ |
$13$ |
[13,0] |
$2^{12}\cdot 3^{6}\cdot 13^{20}$ |
$3$ |
$162.870584734$ |
$204.39719270911772$ |
|
|
? |
$C_{13}:C_6$ (as 13T5) |
trivial |
$2$ |
$12$ |
$95460081074.3$ |
13.13.884...681.1 |
$x^{13} - x^{12} - 144 x^{11} + 161 x^{10} + 6530 x^{9} - 9620 x^{8} - 109398 x^{7} + 196143 x^{6} + 512628 x^{5} - 917970 x^{4} - 650724 x^{3} + 1134730 x^{2} + 253950 x - 409375$ |
$13$ |
[13,0] |
$313^{12}$ |
$1$ |
$201.17749146$ |
$201.177491460457$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$927977436616.0109$ |
13.13.170...729.1 |
$x^{13} - 78 x^{11} + 1989 x^{9} - 1326 x^{8} - 21255 x^{7} + 33813 x^{6} + 68328 x^{5} - 216723 x^{4} + 191178 x^{3} - 51948 x^{2} - 5850 x + 1875$ |
$13$ |
[13,0] |
$3^{12}\cdot 13^{22}$ |
$2$ |
$211.618934626$ |
$249.4371574945215$ |
|
|
? |
$C_{13}:C_6$ (as 13T5) |
trivial |
$2$ |
$12$ |
$394704115709$ |
13.13.116...853.1 |
$x^{13} - 91 x^{11} - 26 x^{10} + 2951 x^{9} + 1339 x^{8} - 41431 x^{7} - 25922 x^{6} + 234988 x^{5} + 213772 x^{4} - 406770 x^{3} - 373113 x^{2} + 223587 x + 170991$ |
$13$ |
[13,0] |
$13^{21}\cdot 19^{6}$ |
$2$ |
$245.268019027$ |
$318.4855383265108$ |
|
|
|
$C_{13}:C_4$ (as 13T4) |
trivial |
$2$ |
$12$ |
$2019070466020$ |
13.13.571...001.1 |
$x^{13} - x^{12} - 204 x^{11} - 181 x^{10} + 10752 x^{9} + 9116 x^{8} - 208418 x^{7} - 161679 x^{6} + 1686466 x^{5} + 1207646 x^{4} - 4904338 x^{3} - 3051848 x^{2} + 896956 x + 144209$ |
$13$ |
[13,0] |
$443^{12}$ |
$1$ |
$277.226159648$ |
$277.22615964805635$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$983737918371.0616$ |
13.13.123...449.1 |
$x^{13} - 3 x^{12} - 278 x^{11} + 205 x^{10} + 24414 x^{9} + 55909 x^{8} - 638959 x^{7} - 3888668 x^{6} - 8795208 x^{5} - 9304040 x^{4} - 4272496 x^{3} - 446336 x^{2} + 199680 x + 41600$ |
$13$ |
[13,0] |
$13^{6}\cdot 131^{12}$ |
$2$ |
$294.1250228866634$ |
$324.62050502705364$ |
|
|
|
$D_{13}$ (as 13T2) |
$[13]$ |
$2$ |
$12$ |
$1357845156620.764$ |
13.13.362...424.1 |
$x^{13} - 214 x^{11} - 528 x^{10} + 15084 x^{9} + 63024 x^{8} - 390420 x^{7} - 2357568 x^{6} + 2053638 x^{5} + 30420288 x^{4} + 30740796 x^{3} - 108639936 x^{2} - 240370524 x - 132080544$ |
$13$ |
[13,0] |
$2^{36}\cdot 3^{16}\cdot 13^{6}\cdot 71^{4}$ |
$4$ |
$319.590116921$ |
$3812.4191094944044$ |
|
|
|
$\PSL(3,3)$ (as 13T7) |
trivial |
$2$ |
$12$ |
$95129277252200$ |
13.13.362...424.2 |
$x^{13} - 222 x^{11} - 64 x^{10} + 18444 x^{9} + 14832 x^{8} - 723404 x^{7} - 1021200 x^{6} + 13437870 x^{5} + 27216624 x^{4} - 96770628 x^{3} - 257555808 x^{2} + 34006548 x + 256489920$ |
$13$ |
[13,0] |
$2^{36}\cdot 3^{16}\cdot 13^{6}\cdot 71^{4}$ |
$4$ |
$319.590116921$ |
$3812.4191094944044$ |
|
|
|
$\PSL(3,3)$ (as 13T7) |
trivial |
$2$ |
$12$ |
$95129277252200$ |
13.13.399...641.1 |
$x^{13} - x^{12} - 240 x^{11} - 293 x^{10} + 19153 x^{9} + 45777 x^{8} - 616830 x^{7} - 1795569 x^{6} + 7791196 x^{5} + 23224049 x^{4} - 29107980 x^{3} - 68466088 x^{2} + 31673025 x + 4516075$ |
$13$ |
[13,0] |
$521^{12}$ |
$1$ |
$321.995799388$ |
$321.99579938787$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$15429359577074.87$ |
13.13.717...241.1 |
$x^{13} - x^{12} - 252 x^{11} + 1123 x^{10} + 15626 x^{9} - 107844 x^{8} - 204415 x^{7} + 3094114 x^{6} - 4853400 x^{5} - 22393129 x^{4} + 91453411 x^{3} - 116380476 x^{2} + 47088126 x - 1165671$ |
$13$ |
[13,0] |
$547^{12}$ |
$1$ |
$336.800646612$ |
$336.80064661199583$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$468882596177901.8$ |
13.13.213...801.1 |
$x^{13} - x^{12} - 276 x^{11} + 1967 x^{10} + 8169 x^{9} - 109375 x^{8} + 114077 x^{7} + 1684091 x^{6} - 4924742 x^{5} - 5465967 x^{4} + 34969245 x^{3} - 20502539 x^{2} - 55304818 x + 57031547$ |
$13$ |
[13,0] |
$599^{12}$ |
$1$ |
$366.250819642$ |
$366.25081964246385$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$1689788670896.741$ |
13.13.926...521.1 |
$x^{13} - x^{12} - 312 x^{11} + 765 x^{10} + 31073 x^{9} - 114643 x^{8} - 1071164 x^{7} + 4472586 x^{6} + 13888428 x^{5} - 61633266 x^{4} - 43862553 x^{3} + 238916059 x^{2} - 140970591 x - 1052321$ |
$13$ |
[13,0] |
$677^{12}$ |
$1$ |
$410.063466568$ |
$410.06346656832307$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$3454123871256.3633$ |
13.13.161...281.1 |
$x^{13} - x^{12} - 396 x^{11} + 1235 x^{10} + 45719 x^{9} - 158783 x^{8} - 2232951 x^{7} + 7665285 x^{6} + 47857178 x^{5} - 162308625 x^{4} - 381260855 x^{3} + 1359880245 x^{2} + 391778734 x - 2211739517$ |
$13$ |
[13,0] |
$859^{12}$ |
$1$ |
$510.859391801$ |
$510.85939180125763$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$107792443172161.95$ |
13.13.326...521.1 |
$x^{13} - x^{12} - 420 x^{11} + 4253 x^{10} + 2721 x^{9} - 193733 x^{8} + 735262 x^{7} + 31458 x^{6} - 3569396 x^{5} + 1482536 x^{4} + 4833237 x^{3} + 1733969 x^{2} - 40719 x - 57869$ |
$13$ |
[13,0] |
$911^{12}$ |
$1$ |
$539.340604042$ |
$539.3406040422938$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$345319680754792.2$ |
13.13.458...681.1 |
$x^{13} - x^{12} - 432 x^{11} + 1203 x^{10} + 46006 x^{9} - 37046 x^{8} - 2039413 x^{7} - 3276218 x^{6} + 27799988 x^{5} + 87214801 x^{4} - 38878963 x^{3} - 420910202 x^{2} - 520002704 x - 190078187$ |
$13$ |
[13,0] |
$937^{12}$ |
$1$ |
$553.533918196$ |
$553.5339181962555$ |
|
✓ |
|
$C_{13}$ (as 13T1) |
trivial |
$2$ |
$12$ |
$169264238822310.44$ |
13.13.308...625.1 |
$x^{13} - 2 x^{12} - 306 x^{11} + 737 x^{10} + 35420 x^{9} - 94473 x^{8} - 1919784 x^{7} + 5276958 x^{6} + 48766779 x^{5} - 126847180 x^{4} - 502327474 x^{3} + 1061716068 x^{2} + 1350962689 x - 1362894158$ |
$13$ |
[13,0] |
$3^{28}\cdot 5^{18}\cdot 29^{12}$ |
$3$ |
$2214.91082149$ |
$16854.804228764082$ |
|
|
? |
$A_{13}$ (as 13T8) |
trivial |
$2$ |
$12$ |
$4344433704670000000$ |