| 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 |
| 8.2.6879707136.2 |
$x^{8} + 6 x^{6} + 9 x^{4} - 2 x^{2} - 9$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{8}$ |
$2$ |
$16.9705627485$ |
$39.41801805957117$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
trivial |
$2$ |
$4$ |
$156.518508673$ |
| 8.2.6879707136.4 |
$x^{8} + 2 x^{6} + 3 x^{4} - 2 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{8}$ |
$2$ |
$16.9705627485$ |
$39.41801805957117$ |
|
|
✓ |
$Q_8:S_4$ (as 8T40) |
trivial |
trivial |
$2$ |
$4$ |
$104.599845355$ |
| 8.2.16384000000.2 |
$x^{8} - 10 x^{6} + 25 x^{4} - 50 x^{2} - 25$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{6}$ |
$2$ |
$18.9148321801$ |
$30.46215435643493$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$148.655297986$ |
| 8.0.20316160000.2 |
$x^{8} + 10 x^{6} + 33 x^{4} + 50 x^{2} + 31$ |
$8$ |
[0,4] |
$2^{20}\cdot 5^{4}\cdot 31$ |
$3$ |
$19.4303311831$ |
$113.42243332841178$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[2]$ |
$[2]$ |
$2$ |
$3$ |
$67.3511830732$ |
| 8.4.20316160000.5 |
$x^{8} - 8 x^{4} - 20 x^{2} + 31$ |
$8$ |
[4,2] |
$2^{20}\cdot 5^{4}\cdot 31$ |
$3$ |
$19.4303311831$ |
$113.42243332841178$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2]$ |
$2$ |
$5$ |
$140.68412796$ |
| 8.0.29132587008.3 |
$x^{8} - 6 x^{6} + 7 x^{4} + 18 x^{2} + 7$ |
$8$ |
[0,4] |
$2^{20}\cdot 3^{4}\cdot 7^{3}$ |
$3$ |
$20.3257875317$ |
$41.748680954623666$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
trivial |
$6$ |
$3$ |
$595.453546445$ |
| 8.0.29132587008.9 |
$x^{8} - 14 x^{6} + 67 x^{4} - 114 x^{2} + 63$ |
$8$ |
[0,4] |
$2^{20}\cdot 3^{4}\cdot 7^{3}$ |
$3$ |
$20.3257875317$ |
$41.748680954623666$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
trivial |
$6$ |
$3$ |
$735.755002751$ |
| 8.2.29948379136.2 |
$x^{8} + 2 x^{6} + 13 x^{4} - 2 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 13^{4}$ |
$2$ |
$20.3960780544$ |
$50.36884737557236$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
trivial |
$2$ |
$4$ |
$161.291152832$ |
| 8.2.29948379136.4 |
$x^{8} - 8 x^{4} + 12 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 13^{4}$ |
$2$ |
$20.3960780544$ |
$50.36884737557236$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
trivial |
$2$ |
$4$ |
$281.439527659$ |
| 8.4.36044800000.3 |
$x^{8} - 2 x^{6} - 29 x^{4} - 30 x^{2} + 55$ |
$8$ |
[4,2] |
$2^{20}\cdot 5^{5}\cdot 11$ |
$3$ |
$20.8739830999$ |
$101.03153630618483$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2]$ |
$2$ |
$5$ |
$299.019650263$ |
| 8.0.36044800000.4 |
$x^{8} + 14 x^{6} + 69 x^{4} + 130 x^{2} + 55$ |
$8$ |
[0,4] |
$2^{20}\cdot 5^{5}\cdot 11$ |
$3$ |
$20.8739830999$ |
$101.03153630618483$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[2]$ |
$[2]$ |
$2$ |
$3$ |
$48.1043849713$ |
| 8.4.51773440000.2 |
$x^{8} + 2 x^{6} - 23 x^{4} - 54 x^{2} + 79$ |
$8$ |
[4,2] |
$2^{20}\cdot 5^{4}\cdot 79$ |
$3$ |
$21.84054029112405$ |
$181.06381179456463$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2]$ |
$2$ |
$5$ |
$335.94733344327403$ |
| 8.2.52144635904.3 |
$x^{8} + 2 x^{6} - 7 x^{4} - 6 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 223^{2}$ |
$2$ |
$21.860052715814263$ |
$136.04592652475563$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$122.10008718109022$ |
| 8.6.52144635904.4 |
$x^{8} - 10 x^{6} + 29 x^{4} - 22 x^{2} - 9$ |
$8$ |
[6,1] |
$-\,2^{20}\cdot 223^{2}$ |
$2$ |
$21.8600527158$ |
$136.04592652475563$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2]$ |
$2$ |
$6$ |
$430.225608747$ |
| 8.2.61917364224.1 |
$x^{8} + 6 x^{6} - 9 x^{4} - 66 x^{2} - 81$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{10}$ |
$2$ |
$22.3345166185$ |
$47.33855458753409$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
$[2]$ |
$2$ |
$4$ |
$374.67125868$ |
| 8.2.61917364224.3 |
$x^{8} - 12 x^{2} - 9$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{10}$ |
$2$ |
$22.3345166185$ |
$47.33855458753409$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
$[2]$ |
$2$ |
$4$ |
$429.621711457$ |
| 8.4.62259200000.1 |
$x^{8} - 18 x^{6} + 111 x^{4} - 250 x^{2} + 95$ |
$8$ |
[4,2] |
$2^{20}\cdot 5^{5}\cdot 19$ |
$3$ |
$22.3498926757$ |
$132.78145244223714$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2]$ |
$2$ |
$5$ |
$415.760409466$ |
| 8.0.62259200000.2 |
$x^{8} + 4 x^{6} - 56 x^{4} + 80 x^{2} + 95$ |
$8$ |
[0,4] |
$2^{20}\cdot 5^{5}\cdot 19$ |
$3$ |
$22.3498926757$ |
$132.78145244223714$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[2]$ |
$[2]$ |
$2$ |
$3$ |
$55.5276162936$ |
| 8.2.79298560000.1 |
$x^{8} - 2 x^{6} - 13 x^{4} - 286 x^{2} - 121$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 11^{2}$ |
$3$ |
$23.0360579626$ |
$67.5638604016345$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$241.993479894$ |
| 8.2.79298560000.3 |
$x^{8} - 14 x^{6} + 63 x^{4} - 198 x^{2} - 121$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 11^{2}$ |
$3$ |
$23.0360579626$ |
$67.5638604016345$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$118.378393838$ |
| 8.2.110755840000.4 |
$x^{8} - 14 x^{6} + 31 x^{4} + 26 x^{2} - 169$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 13^{2}$ |
$3$ |
$24.0184971236$ |
$73.44965995393171$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$447.219364429$ |
| 8.2.110755840000.6 |
$x^{8} - 2 x^{6} - 31 x^{4} + 182 x^{2} - 169$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 13^{2}$ |
$3$ |
$24.0184971236$ |
$73.44965995393171$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$238.235697454$ |
| 8.0.162310127616.4 |
$x^{8} - 2 x^{6} + 145 x^{4} + 150 x^{2} + 39$ |
$8$ |
[0,4] |
$2^{20}\cdot 3^{5}\cdot 7^{2}\cdot 13$ |
$4$ |
$25.193772444633453$ |
$198.10468593061537$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
trivial |
$6$ |
$3$ |
$2946.119658150921$ |
| 8.0.162310127616.6 |
$x^{8} + 6 x^{6} + 39 x^{4} + 126 x^{2} + 351$ |
$8$ |
[0,4] |
$2^{20}\cdot 3^{5}\cdot 7^{2}\cdot 13$ |
$4$ |
$25.193772444633453$ |
$198.10468593061537$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
trivial |
$6$ |
$3$ |
$1471.3065678093667$ |
| 8.2.189399040000.3 |
$x^{8} - 14 x^{6} + 61 x^{4} - 34 x^{2} - 289$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 17^{2}$ |
$3$ |
$25.6845654061$ |
$83.99289956479353$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$590.430073209$ |
| 8.2.189399040000.6 |
$x^{8} + 2 x^{6} - 61 x^{4} + 238 x^{2} - 289$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 17^{2}$ |
$3$ |
$25.6845654061$ |
$83.99289956479353$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$274.511348449$ |
| 8.2.236584960000.2 |
$x^{8} - 4 x^{6} - 32 x^{4} - 608 x^{2} - 361$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 19^{2}$ |
$3$ |
$26.4087832163$ |
$88.79630900143394$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$328.821239939$ |
| 8.2.236584960000.4 |
$x^{8} - 18 x^{6} + 117 x^{4} - 494 x^{2} - 361$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 19^{2}$ |
$3$ |
$26.4087832163$ |
$88.79630900143394$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$191.020616857$ |
| 8.0.262193283072.18 |
$x^{8} + 18 x^{6} + 99 x^{4} + 126 x^{2} + 63$ |
$8$ |
[0,4] |
$2^{20}\cdot 3^{6}\cdot 7^{3}$ |
$3$ |
$26.7502407632$ |
$89.37119854321661$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$6$ |
$3$ |
$453.662113736$ |
| 8.0.262193283072.20 |
$x^{8} - 18 x^{6} + 129 x^{4} - 378 x^{2} + 567$ |
$8$ |
[0,4] |
$2^{20}\cdot 3^{6}\cdot 7^{3}$ |
$3$ |
$26.7502407632$ |
$89.37119854321661$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$6$ |
$3$ |
$936.664171754$ |
| 8.2.269535412224.1 |
$x^{8} + 2 x^{6} - 13 x^{4} - 66 x^{2} - 81$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{2}\cdot 13^{4}$ |
$3$ |
$26.8427482935$ |
$56.893861957235075$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$364.986147913$ |
| 8.2.269535412224.3 |
$x^{8} - 10 x^{6} + 39 x^{4} - 18 x^{2} - 81$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{2}\cdot 13^{4}$ |
$3$ |
$26.8427482935$ |
$56.893861957235075$ |
|
|
|
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1703.28987187$ |
| 8.0.353100627968.3 |
$x^{8} + 6 x^{6} - 16 x^{5} + 25 x^{4} - 48 x^{3} + 46 x^{2} - 40 x + 31$ |
$8$ |
[0,4] |
$2^{20}\cdot 11^{4}\cdot 23$ |
$3$ |
$27.76433726$ |
$144.90833678194346$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
trivial |
$2$ |
$3$ |
$455.818939068$ |
| 8.2.551157760000.8 |
$x^{8} - 6 x^{6} - 57 x^{4} - 1102 x^{2} - 841$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 29^{2}$ |
$3$ |
$29.3534728634$ |
$109.70264840541167$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$464.606475747$ |
| 8.2.551157760000.10 |
$x^{8} + 22 x^{6} + 187 x^{4} + 986 x^{2} - 841$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 29^{2}$ |
$3$ |
$29.3534728634$ |
$109.70264840541167$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$279.906721908$ |
| 8.2.629800960000.4 |
$x^{8} + 10 x^{6} + 11 x^{4} - 90 x^{2} - 81$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 31^{2}$ |
$3$ |
$29.8469813893$ |
$113.42243332841178$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$254.834244204$ |
| 8.6.629800960000.8 |
$x^{8} + 2 x^{6} - 45 x^{4} + 14 x^{2} - 1$ |
$8$ |
[6,1] |
$-\,2^{20}\cdot 5^{4}\cdot 31^{2}$ |
$3$ |
$29.8469813893$ |
$113.42243332841178$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2]$ |
$2$ |
$6$ |
$1520.18234472$ |
| 8.2.748709478400.5 |
$x^{8} - 10 x^{6} - 3 x^{4} + 10 x^{2} - 25$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{2}\cdot 13^{4}$ |
$3$ |
$30.4992504603$ |
$73.44965995393171$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1455.821807$ |
| 8.2.748709478400.7 |
$x^{8} - 2 x^{6} + 3 x^{4} + 50 x^{2} - 25$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{2}\cdot 13^{4}$ |
$3$ |
$30.4992504603$ |
$73.44965995393171$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1677.70955479$ |
| 8.6.886458810368.2 |
$x^{8} - 16 x^{6} + 8 x^{4} + 60 x^{2} - 17$ |
$8$ |
[6,1] |
$-\,2^{20}\cdot 17\cdot 223^{2}$ |
$3$ |
$31.1499475758$ |
$560.9317249965869$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$[2, 2]$ |
$2$ |
$6$ |
$3075.70992048$ |
| 8.4.1167986786304.4 |
$x^{8} - 2 x^{6} + x^{4} - 26 x^{2} + 39$ |
$8$ |
[4,2] |
$2^{20}\cdot 3\cdot 13^{5}$ |
$3$ |
$32.2425766676$ |
$108.03171057526986$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2]$ |
$2$ |
$5$ |
$2024.01608538$ |
| 8.0.1167986786304.9 |
$x^{8} + 16 x^{6} + 64 x^{4} + 52 x^{2} + 39$ |
$8$ |
[0,4] |
$2^{20}\cdot 3\cdot 13^{5}$ |
$3$ |
$32.2425766676$ |
$108.03171057526986$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[2]$ |
$[2]$ |
$2$ |
$3$ |
$369.44786626$ |
| 8.2.1327104000000.5 |
$x^{8} + 30 x^{6} + 225 x^{4} + 1350 x^{2} - 2025$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{4}\cdot 5^{6}$ |
$3$ |
$32.7614503525$ |
$52.76199905335091$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2, 2]$ |
$2$ |
$4$ |
$475.276607474$ |
| 8.4.1645608960000.1 |
$x^{8} - 30 x^{6} + 297 x^{4} - 1350 x^{2} + 2511$ |
$8$ |
[4,2] |
$2^{20}\cdot 3^{4}\cdot 5^{4}\cdot 31$ |
$4$ |
$33.654320816945216$ |
$196.45341724290276$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$[2, 2]$ |
$2$ |
$5$ |
$1248.5549010495188$ |
| 8.2.1840906240000.2 |
$x^{8} - 10 x^{6} + 73 x^{4} - 530 x^{2} - 2809$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 53^{2}$ |
$3$ |
$34.1294239958$ |
$148.30508705665198$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1241.0505782$ |
| 8.2.1840906240000.3 |
$x^{8} + 10 x^{6} - 73 x^{4} + 530 x^{2} - 2809$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 53^{2}$ |
$3$ |
$34.1294239958$ |
$148.30508705665198$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$811.612647078$ |
| 8.2.2281308160000.7 |
$x^{8} + 32 x^{6} + 352 x^{4} + 2596 x^{2} - 3481$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 59^{2}$ |
$3$ |
$35.0568583826$ |
$156.47469696986494$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1900.95563488$ |
| 8.2.2281308160000.8 |
$x^{8} + 6 x^{6} - 87 x^{4} + 3422 x^{2} - 3481$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 5^{4}\cdot 59^{2}$ |
$3$ |
$35.0568583826$ |
$156.47469696986494$ |
|
|
? |
$(C_4^2 : C_2):C_2$ (as 8T26) |
$[2]$ |
$[2]$ |
$2$ |
$4$ |
$630.655722952$ |
| 8.2.2425818710016.1 |
$x^{8} - 72 x^{4} - 324 x^{2} - 81$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{4}\cdot 13^{4}$ |
$3$ |
$35.3270434653$ |
$87.24140277317363$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1503.92037545$ |
| 8.2.2425818710016.9 |
$x^{8} - 6 x^{6} + 117 x^{4} + 54 x^{2} - 81$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 3^{4}\cdot 13^{4}$ |
$3$ |
$35.3270434653$ |
$87.24140277317363$ |
|
|
? |
$Q_8:S_4$ (as 8T40) |
trivial |
$[2]$ |
$2$ |
$4$ |
$1636.47341098$ |
