| Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Narrow class group |
Unit group torsion |
Unit group rank |
Regulator |
| 8.2.1354432512.4 |
$x^{8} - 2 x^{6} - 2 x^{4} + 2 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3\cdot 83^{2}$ |
$3$ |
$13.8506345733$ |
$89.26365441768559$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$4$ |
$39.1242281751$ |
| 8.2.1354432512.6 |
$x^{8} - 2 x^{6} - 3 x^{4} + 8 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3\cdot 83^{2}$ |
$3$ |
$13.8506345733$ |
$89.26365441768559$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$4$ |
$31.4149011924$ |
| 8.4.5120000000.1 |
$x^{8} - 15 x^{4} - 10 x^{2} + 5$ |
$8$ |
[4,2] |
$2^{16}\cdot 5^{7}$ |
$2$ |
$16.3553086792$ |
$23.129899350864836$ |
|
|
? |
$C_8:C_2$ (as 8T7) |
trivial |
$[2]$ |
$2$ |
$5$ |
$88.2974472278$ |
| 8.2.12189892608.4 |
$x^{8} + 2 x^{6} + 6 x^{4} - 18 x^{2} - 27$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{3}\cdot 83^{2}$ |
$3$ |
$18.2284602249$ |
$89.26365441768559$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$4$ |
$180.79917918$ |
| 8.2.12189892608.5 |
$x^{8} + 8 x^{6} + 9 x^{4} - 18 x^{2} - 27$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{3}\cdot 83^{2}$ |
$3$ |
$18.2284602249$ |
$89.26365441768559$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$4$ |
$148.172356231$ |
| 8.0.15909322752.2 |
$x^{8} - 6 x^{6} - 4 x^{5} + 18 x^{4} + 4 x^{2} - 24 x + 12$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{8}\cdot 37$ |
$3$ |
$18.845447803$ |
$123.9702037636292$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$99.3217886281$ |
| 8.0.18839568384.6 |
$x^{8} + 10 x^{6} + 37 x^{4} + 48 x^{2} + 21$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{5}\cdot 7\cdot 13^{2}$ |
$4$ |
$19.2479228215$ |
$123.00892591564936$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$176.37870231$ |
| 8.0.18839568384.8 |
$x^{8} - 6 x^{6} + 30 x^{4} - 90 x^{2} + 189$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{5}\cdot 7\cdot 13^{2}$ |
$4$ |
$19.2479228215$ |
$123.00892591564936$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$122.168784446$ |
| 8.6.19456000000.1 |
$x^{8} - 14 x^{6} + 46 x^{4} - 34 x^{2} - 19$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 19$ |
$3$ |
$19.3255425254$ |
$82.447842006928$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$279.03356717$ |
| 8.2.19456000000.6 |
$x^{8} + 2 x^{6} - 11 x^{4} - 32 x^{2} - 19$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{6}\cdot 19$ |
$3$ |
$19.3255425254$ |
$82.447842006928$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$4$ |
$77.6399847521$ |
| 8.6.22077440000.1 |
$x^{8} - 10 x^{6} - 4 x^{5} + 22 x^{4} - 36 x^{2} - 24 x - 4$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{4}\cdot 7^{2}\cdot 11$ |
$4$ |
$19.6333129483$ |
$110.99549540409286$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$[2]$ |
$2$ |
$6$ |
$289.740748627$ |
| 8.4.23928307712.1 |
$x^{8} - 8 x^{6} + 41 x^{4} - 98 x^{2} + 53$ |
$8$ |
[4,2] |
$2^{16}\cdot 53\cdot 83^{2}$ |
$3$ |
$19.8318851241$ |
$375.19061821959247$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$5$ |
$180.306742339$ |
| 8.4.27540127744.1 |
$x^{8} - 8 x^{6} + 41 x^{4} - 102 x^{2} + 61$ |
$8$ |
[4,2] |
$2^{16}\cdot 61\cdot 83^{2}$ |
$3$ |
$20.1834658325$ |
$402.512111619017$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$5$ |
$220.692492874$ |
| 8.4.29696000000.4 |
$x^{8} - 8 x^{6} - 8 x^{5} + 5 x^{4} + 32 x^{3} + 30 x^{2} - 36 x - 11$ |
$8$ |
[4,2] |
$2^{16}\cdot 5^{6}\cdot 29$ |
$3$ |
$20.3745133503$ |
$116.47936428959294$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$[2]$ |
$2$ |
$5$ |
$242.373075063$ |
| 8.0.30433148928.4 |
$x^{8} + 8 x^{6} + 33 x^{4} + 38 x^{2} + 13$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{6}\cdot 7^{2}\cdot 13$ |
$4$ |
$20.437057245$ |
$123.00892591564936$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$134.701794308$ |
| 8.0.30433148928.6 |
$x^{8} + 6 x^{6} - 3 x^{4} - 72 x^{2} + 117$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{6}\cdot 7^{2}\cdot 13$ |
$4$ |
$20.437057245$ |
$123.00892591564936$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$194.472980856$ |
| 8.4.34763767808.1 |
$x^{8} - 14 x^{6} + 62 x^{4} - 114 x^{2} + 77$ |
$8$ |
[4,2] |
$2^{16}\cdot 7\cdot 11\cdot 83^{2}$ |
$4$ |
$20.7797756838$ |
$452.23002996262863$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$5$ |
$390.620828688$ |
| 8.4.37930467328.3 |
$x^{8} - 6 x^{6} + 13 x^{4} - 16 x^{2} + 13$ |
$8$ |
[4,2] |
$2^{16}\cdot 13\cdot 211^{2}$ |
$3$ |
$21.0074589798$ |
$296.2701469942593$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$5$ |
$272.865988608$ |
| 8.0.49161240576.3 |
$x^{8} + 9 x^{4} + 30 x^{2} + 21$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{7}\cdot 7^{3}$ |
$3$ |
$21.6996562542$ |
$39.138583805161$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$152.989319019$ |
| 8.0.49161240576.7 |
$x^{8} + 36 x^{4} - 240 x^{2} + 336$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{7}\cdot 7^{3}$ |
$3$ |
$21.6996562542$ |
$39.138583805161$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$152.989319019$ |
| 8.2.50537889792.2 |
$x^{8} + 6 x^{6} + 6 x^{4} - 22 x^{2} - 27$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{3}\cdot 13^{4}$ |
$3$ |
$21.7746978819$ |
$35.32704346531139$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$[2]$ |
$2$ |
$4$ |
$818.264124546$ |
| 8.2.50537889792.5 |
$x^{8} - 6 x^{6} - 18 x^{4} - 10 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{3}\cdot 13^{4}$ |
$3$ |
$21.7746978819$ |
$35.32704346531139$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$[2]$ |
$2$ |
$4$ |
$818.264124546$ |
| 8.2.54517760000.7 |
$x^{8} + 2 x^{6} - 10 x^{4} + 14 x^{2} - 11$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{4}\cdot 11^{3}$ |
$3$ |
$21.9820023738$ |
$41.95235392680606$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$[2]$ |
$2$ |
$4$ |
$212.462358706$ |
| 8.2.54517760000.8 |
$x^{8} + 6 x^{6} - 35 x^{4} + 48 x^{2} - 11$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{4}\cdot 11^{3}$ |
$3$ |
$21.9820023738$ |
$41.95235392680606$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$[2]$ |
$2$ |
$4$ |
$212.462358706$ |
| 8.6.60416000000.4 |
$x^{8} - 16 x^{6} + 41 x^{4} + 94 x^{2} - 59$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 59$ |
$3$ |
$22.2660918391$ |
$145.28758277154319$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$445.725260616$ |
| 8.2.60416000000.7 |
$x^{8} + 8 x^{6} + 9 x^{4} - 38 x^{2} - 59$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{6}\cdot 59$ |
$3$ |
$22.2660918391$ |
$145.28758277154319$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$4$ |
$104.106061519$ |
| 8.0.62464000000.1 |
$x^{8} + 14 x^{6} + 61 x^{4} + 104 x^{2} + 61$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{6}\cdot 61$ |
$3$ |
$22.3590694002$ |
$147.72956190416983$ |
✓ |
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
$[10]$ |
$[10]$ |
$2$ |
$3$ |
$14.8224845118$ |
| 8.8.62464000000.1 |
$x^{8} - 18 x^{6} + 109 x^{4} - 232 x^{2} + 61$ |
$8$ |
[8,0] |
$2^{16}\cdot 5^{6}\cdot 61$ |
$3$ |
$22.3590694002$ |
$147.72956190416983$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$7$ |
$540.705376658$ |
| 8.4.73932800000.3 |
$x^{8} - 2 x^{6} + 6 x^{4} - 30 x^{2} + 5$ |
$8$ |
[4,2] |
$2^{16}\cdot 5^{5}\cdot 19^{2}$ |
$3$ |
$22.8351907769$ |
$82.447842006928$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) |
trivial |
$[2]$ |
$2$ |
$5$ |
$549.59438667$ |
| 8.0.73932800000.9 |
$x^{8} + 14 x^{6} + 69 x^{4} + 120 x^{2} + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{5}\cdot 19^{2}$ |
$3$ |
$22.8351907769$ |
$82.447842006928$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) |
$[2]$ |
$[2]$ |
$2$ |
$3$ |
$76.4612305138$ |
| 8.4.99204005888.2 |
$x^{8} - 2 x^{6} - 8 x^{5} - 14 x^{4} + 8 x^{3} + 18 x^{2} + 8 x + 1$ |
$8$ |
[4,2] |
$2^{16}\cdot 13^{4}\cdot 53$ |
$3$ |
$23.6900594827$ |
$148.48568954616468$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$[2]$ |
$2$ |
$5$ |
$464.680672116$ |
| 8.0.104963309568.4 |
$x^{8} + 24 x^{6} + 201 x^{4} + 702 x^{2} + 1053$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{6}\cdot 13^{3}$ |
$3$ |
$23.8577613212$ |
$88.28226040378215$ |
|
|
? |
$C_4\wr C_2$ (as 8T17) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$358.401088438$ |
| 8.0.104963309568.13 |
$x^{8} - 24 x^{6} + 177 x^{4} - 234 x^{2} + 117$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{6}\cdot 13^{3}$ |
$3$ |
$23.8577613212$ |
$88.28226040378215$ |
|
|
? |
$C_4\wr C_2$ (as 8T17) |
$[4]$ |
$[4]$ |
$6$ |
$3$ |
$188.059882869$ |
| 8.6.142336000000.2 |
$x^{8} - 18 x^{6} + 69 x^{4} + 8 x^{2} - 139$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 139$ |
$3$ |
$24.7835973478$ |
$223.0025825401924$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$475.04788293$ |
| 8.6.183296000000.6 |
$x^{8} - 8 x^{6} - 31 x^{4} + 278 x^{2} - 179$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 179$ |
$3$ |
$25.5796220589$ |
$253.06320727358604$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$636.985086686$ |
| 8.4.209920000000.7 |
$x^{8} - 55 x^{4} + 90 x^{2} + 205$ |
$8$ |
[4,2] |
$2^{16}\cdot 5^{7}\cdot 41$ |
$3$ |
$26.0169706613$ |
$148.10361914290496$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2, 2]$ |
$2$ |
$5$ |
$439.551078775$ |
| 8.4.209920000000.8 |
$x^{8} - 10 x^{6} - 50 x^{4} + 90 x^{2} + 205$ |
$8$ |
[4,2] |
$2^{16}\cdot 5^{7}\cdot 41$ |
$3$ |
$26.0169706613$ |
$148.10361914290496$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2, 2]$ |
$2$ |
$5$ |
$425.692914929$ |
| 8.2.230599360512.5 |
$x^{8} - 2 x^{6} + 13 x^{4} - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{3}\cdot 19^{4}$ |
$3$ |
$26.3243261132$ |
$91.81599200896666$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
$[2]$ |
$2$ |
$4$ |
$708.00813029$ |
| 8.2.230599360512.7 |
$x^{8} - 6 x^{6} - 2 x^{4} + 6 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{3}\cdot 19^{4}$ |
$3$ |
$26.3243261132$ |
$91.81599200896666$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
$[2]$ |
$2$ |
$4$ |
$399.416841607$ |
| 8.6.388096000000.2 |
$x^{8} - 18 x^{6} + 54 x^{4} + 98 x^{2} - 379$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 379$ |
$3$ |
$28.0942469662$ |
$368.2324838408302$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$850.427793453$ |
| 8.6.404480000000.2 |
$x^{8} - 10 x^{6} - 35 x^{4} + 360 x^{2} - 395$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{7}\cdot 79$ |
$3$ |
$28.2398334814$ |
$205.5830422834283$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2, 2]$ |
$2$ |
$6$ |
$1111.45477213$ |
| 8.4.414720000000.3 |
$x^{8} - 135 x^{4} + 270 x^{2} + 405$ |
$8$ |
[4,2] |
$2^{16}\cdot 3^{4}\cdot 5^{7}$ |
$3$ |
$28.3282256058$ |
$40.062160849652294$ |
|
|
? |
$C_8:C_2$ (as 8T7) |
$[2]$ |
$[2, 2]$ |
$2$ |
$5$ |
$403.746424871$ |
| 8.6.429056000000.6 |
$x^{8} + 6 x^{6} - 139 x^{4} + 456 x^{2} - 419$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 419$ |
$3$ |
$28.4488193456$ |
$387.17695852360055$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$1407.04511677$ |
| 8.2.490659840000.7 |
$x^{8} - 16 x^{6} + 41 x^{4} - 66 x^{2} - 99$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{2}\cdot 5^{4}\cdot 11^{3}$ |
$4$ |
$28.9299420769$ |
$132.33207809551135$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$581.301712934$ |
| 8.2.490659840000.8 |
$x^{8} - 14 x^{6} + 62 x^{4} - 66 x^{2} - 99$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{2}\cdot 5^{4}\cdot 11^{3}$ |
$4$ |
$28.9299420769$ |
$132.33207809551135$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$703.649106206$ |
| 8.6.510976000000.6 |
$x^{8} - 22 x^{6} + 29 x^{4} + 952 x^{2} - 499$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 499$ |
$3$ |
$29.0770312217$ |
$422.52534518485777$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$813.266917494$ |
| 8.8.619520000000.2 |
$x^{8} - 30 x^{6} + 270 x^{4} - 770 x^{2} + 605$ |
$8$ |
[8,0] |
$2^{16}\cdot 5^{7}\cdot 11^{2}$ |
$3$ |
$29.7856386455$ |
$76.7131975855036$ |
|
|
? |
$C_8:C_2$ (as 8T7) |
trivial |
$[2, 2]$ |
$2$ |
$7$ |
$2192.09929612$ |
| 8.0.619520000000.4 |
$x^{8} + 30 x^{6} + 285 x^{4} + 880 x^{2} + 605$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{7}\cdot 11^{2}$ |
$3$ |
$29.7856386455$ |
$76.7131975855036$ |
✓ |
|
? |
$C_8:C_2$ (as 8T7) |
$[2, 10]$ |
$[2, 10]$ |
$2$ |
$3$ |
$14.8224845118$ |
| 8.6.633856000000.5 |
$x^{8} - 8 x^{6} - 151 x^{4} + 638 x^{2} - 619$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 619$ |
$3$ |
$29.8709356118$ |
$470.5955508624993$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$995.817714721$ |
| 8.6.674816000000.3 |
$x^{8} - 24 x^{6} + 121 x^{4} + 266 x^{2} - 659$ |
$8$ |
[6,1] |
$-\,2^{16}\cdot 5^{6}\cdot 659$ |
$3$ |
$30.105661304$ |
$485.56256810793997$ |
|
|
? |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$[2]$ |
$2$ |
$6$ |
$1667.53942813$ |
