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 |
8.0.47775744.1 |
$x^{8} + 3 x^{4} + 9$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{6}$ |
$2$ |
$9.11802822782$ |
$9.11802822781911$ |
|
✓ |
? |
$D_4$ (as 8T4) |
trivial |
$12$ |
$3$ |
$20.8957982344$ |
8.0.55115776.3 |
$x^{8} - 4 x^{5} - x^{4} + 4 x^{3} + 8 x^{2} + 4 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 29^{2}$ |
$2$ |
$9.28238314842$ |
$21.540659228538015$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$8.52019638775$ |
8.0.321978368.4 |
$x^{8} - 4 x^{6} + 11 x^{4} - 20 x^{2} + 17$ |
$8$ |
[0,4] |
$2^{16}\cdot 17^{3}$ |
$2$ |
$11.5738629877$ |
$16.492422502470642$ |
|
|
? |
$D_{8}$ (as 8T6) |
trivial |
$4$ |
$3$ |
$20.282622579$ |
8.0.375455744.1 |
$x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 22 x^{4} - 16 x^{3} + 20 x^{2} - 8 x + 4$ |
$8$ |
[0,4] |
$2^{16}\cdot 17\cdot 337$ |
$3$ |
$11.7983111155$ |
$302.76063152266016$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$22.7577294403$ |
8.0.430505984.1 |
$x^{8} - 4 x^{7} + 12 x^{6} - 20 x^{5} + 23 x^{4} - 20 x^{3} + 20 x^{2} - 8 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 6569$ |
$2$ |
$12.0018280145$ |
$324.1974706872341$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$25.0864433646$ |
8.0.451477504.5 |
$x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 21 x^{4} - 20 x^{3} + 20 x^{2} - 12 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 83^{2}$ |
$2$ |
$12.0733979172$ |
$36.4417343165772$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$35.694160127$ |
8.0.481361920.1 |
$x^{8} + 2 x^{6} - 8 x^{5} + x^{4} - 12 x^{3} + 16 x^{2} + 13$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 13\cdot 113$ |
$4$ |
$12.1705152911$ |
$342.81190177705327$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$29.9300203392$ |
8.0.514916352.6 |
$x^{8} - 8 x^{6} + 2 x^{4} + 56 x^{2} + 97$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 97$ |
$3$ |
$12.273462089$ |
$68.23488843692792$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) |
trivial |
$12$ |
$3$ |
$53.6338974265$ |
8.0.536412160.1 |
$x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 14 x^{4} - 4 x^{3} - 10 x^{2} + 8 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 1637$ |
$3$ |
$12.3363683032$ |
$361.88395930187346$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$34.747638937$ |
8.0.642318336.9 |
$x^{8} - 12 x^{6} + 59 x^{4} - 132 x^{2} + 121$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 11^{2}$ |
$3$ |
$12.6173685821$ |
$22.978250586152114$ |
|
|
? |
$D_4\times C_2$ (as 8T9) |
trivial |
$12$ |
$3$ |
$85.4809876926$ |
8.0.750321664.3 |
$x^{8} - 6 x^{6} - 4 x^{5} + 14 x^{4} + 32 x^{3} - 34 x^{2} - 60 x + 61$ |
$8$ |
[0,4] |
$2^{16}\cdot 107^{2}$ |
$2$ |
$12.8648858108$ |
$41.376321731154405$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$29.8794137621$ |
8.0.881917952.2 |
$x^{8} - 2 x^{6} - 8 x^{5} + 14 x^{4} - 4 x^{2} + 8 x + 4$ |
$8$ |
[0,4] |
$2^{16}\cdot 13457$ |
$2$ |
$13.127394082$ |
$464.0172410589934$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$45.7293913147$ |
8.0.897122304.10 |
$x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 13^{2}$ |
$3$ |
$13.1554726719$ |
$24.979991993593593$ |
✓ |
|
? |
$D_4\times C_2$ (as 8T9) |
trivial |
$12$ |
$3$ |
$80.5956958474$ |
8.0.921763840.1 |
$x^{8} - 4 x^{7} + 6 x^{6} - 14 x^{4} + 16 x^{3} + 4 x^{2} - 24 x + 20$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 29\cdot 97$ |
$4$ |
$13.2001072061$ |
$474.3838108536167$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$47.8521365508$ |
8.0.1024000000.6 |
$x^{8} + 4 x^{6} + 16 x^{4} - 56 x^{2} + 36$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{6}$ |
$2$ |
$13.3748060995$ |
$13.37480609952844$ |
|
✓ |
|
$D_4$ (as 8T4) |
$[2]$ |
$4$ |
$3$ |
$24.6646630038$ |
8.0.1145110528.2 |
$x^{8} - 4 x^{6} + 9 x^{4} - 4 x^{3} - 2 x^{2} - 4 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 101\cdot 173$ |
$3$ |
$13.5630051276$ |
$528.7419030112897$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$37.7270188141$ |
8.0.1174470656.1 |
$x^{8} + 4 x^{6} - 4 x^{5} + x^{4} - 8 x^{3} + 2 x^{2} + 8 x + 13$ |
$8$ |
[0,4] |
$2^{16}\cdot 17921$ |
$2$ |
$13.6059938809$ |
$535.4773571309995$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$43.7260685499$ |
8.0.1266221056.1 |
$x^{8} - 4 x^{6} - 4 x^{5} + 25 x^{4} + 8 x^{3} - 38 x^{2} - 24 x + 45$ |
$8$ |
[0,4] |
$2^{16}\cdot 139^{2}$ |
$2$ |
$13.7345264921$ |
$47.15930449020638$ |
|
|
|
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$55.5478932997$ |
8.0.1364787200.1 |
$x^{8} + 2 x^{6} - 12 x^{5} + 22 x^{4} - 16 x^{3} + 6 x^{2} - 4 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{2}\cdot 7^{2}\cdot 17$ |
$4$ |
$13.8638265914$ |
$127.58907089700965$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$54.2440615096$ |
8.0.1382612992.4 |
$x^{8} - 4 x^{6} + 11 x^{4} + 4 x^{2} + 73$ |
$8$ |
[0,4] |
$2^{16}\cdot 17^{2}\cdot 73$ |
$3$ |
$13.8863331145$ |
$140.91131963046828$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$4$ |
$3$ |
$63.3566161084$ |
8.0.1506344960.1 |
$x^{8} - 4 x^{5} + 5 x^{4} + 2 x^{2} - 8 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 4597$ |
$3$ |
$14.0359093548$ |
$606.4321891192782$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$75.0156014077$ |
8.0.1560346624.1 |
$x^{8} + 4 x^{6} - 4 x^{5} + 15 x^{4} - 20 x^{3} + 12 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 29\cdot 821$ |
$3$ |
$14.0978418946$ |
$617.2066104636275$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$55.7849224907$ |
8.0.1615396864.3 |
$x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 13 x^{4} - 12 x^{3} + 6 x^{2} - 8 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 157^{2}$ |
$2$ |
$14.1590757247$ |
$50.11985634456667$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$73.2862051196$ |
8.0.1629552640.1 |
$x^{8} - 4 x^{7} + 10 x^{6} - 8 x^{5} - 2 x^{4} + 20 x^{3} - 10 x^{2} - 8 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 4973$ |
$3$ |
$14.1745261433$ |
$630.7455905513727$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$52.8989617706$ |
8.0.1637941248.2 |
$x^{8} + 4 x^{6} + x^{4} - 4 x^{3} - 2 x^{2} - 4 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{2}\cdot 2777$ |
$3$ |
$14.1836266172$ |
$365.09724731912183$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$61.0440450374$ |
8.0.1698234368.3 |
$x^{8} - 4 x^{7} + 16 x^{6} - 32 x^{5} + 59 x^{4} - 56 x^{3} + 48 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 25913$ |
$2$ |
$14.2478621038$ |
$643.9006134490012$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$54.2092673956$ |
8.2.1863254016.1 |
$x^{8} - 6 x^{6} - 12 x^{5} - 18 x^{4} - 24 x^{3} - 18 x^{2} - 12 x - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{7}\cdot 13$ |
$3$ |
$14.4139830907$ |
$37.71488305085648$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$4$ |
$45.6290093489$ |
8.0.1916338176.6 |
$x^{8} - 37 x^{4} + 361$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 19^{2}$ |
$3$ |
$14.4646862838$ |
$30.199337741083$ |
|
|
? |
$D_4\times C_2$ (as 8T9) |
trivial |
$12$ |
$3$ |
$150.856250519$ |
8.0.2062614528.1 |
$x^{8} + 4 x^{6} + 9 x^{4} - 36 x^{3} + 46 x^{2} - 36 x + 13$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{2}\cdot 13\cdot 269$ |
$4$ |
$14.598299151$ |
$409.70233096725235$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$55.8690749893$ |
8.0.2096168960.1 |
$x^{8} - 4 x^{7} + 2 x^{6} + 14 x^{4} - 4 x^{3} - 2 x^{2} - 8 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 6397$ |
$3$ |
$14.6277754775$ |
$715.3740280440715$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$75.6061907174$ |
8.0.2163802112.1 |
$x^{8} - 4 x^{7} + 6 x^{6} - 6 x^{4} + 8 x^{3} + 12 x^{2} - 24 x + 20$ |
$8$ |
[0,4] |
$2^{16}\cdot 137\cdot 241$ |
$3$ |
$14.6859549802$ |
$726.8232247252422$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$51.1032575152$ |
8.0.2298544128.3 |
$x^{8} - 4 x^{6} - 21 x^{4} + 140 x^{2} + 433$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 433$ |
$3$ |
$14.7972702539$ |
$144.1665703275208$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) |
trivial |
$12$ |
$3$ |
$157.057924835$ |
8.0.2341011456.19 |
$x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 11 x^{4} - 16 x^{3} + 8 x^{2} + 4 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{6}\cdot 7^{2}$ |
$3$ |
$14.8311710043$ |
$38.12856628395392$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$58.704350859$ |
8.0.2357264384.1 |
$x^{8} - 4 x^{7} + 2 x^{6} + 8 x^{5} - 10 x^{4} - 4 x^{3} + 30 x^{2} - 32 x + 13$ |
$8$ |
[0,4] |
$2^{16}\cdot 35969$ |
$2$ |
$14.8440031149$ |
$758.6197993725184$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$55.9164140875$ |
8.0.2553348096.6 |
$x^{8} - 16 x^{6} + 107 x^{4} - 344 x^{2} + 481$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 13\cdot 37$ |
$4$ |
$14.9930073271$ |
$151.94735930578062$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) |
trivial |
$12$ |
$3$ |
$149.879530135$ |
8.0.2636185600.1 |
$x^{8} + 2 x^{6} - 8 x^{5} + 2 x^{4} - 8 x^{3} + 36 x^{2} + 24 x + 52$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{2}\cdot 1609$ |
$3$ |
$15.0529631311$ |
$469.1568009271788$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$72.8785050392$ |
8.0.2723217408.3 |
$x^{8} - 6 x^{6} - 12 x^{5} + 6 x^{4} + 48 x^{3} + 78 x^{2} + 60 x + 21$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{7}\cdot 19$ |
$3$ |
$15.1142043351$ |
$45.595070303079446$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
$[2]$ |
$6$ |
$3$ |
$61.2644178686$ |
8.0.2746286080.3 |
$x^{8} - 8 x^{6} + 34 x^{4} - 104 x^{2} + 145$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 17^{2}\cdot 29$ |
$4$ |
$15.1301495931$ |
$198.59506539690256$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$4$ |
$3$ |
$54.4782307065$ |
8.0.2775646208.2 |
$x^{8} - 4 x^{6} - 4 x^{5} + 7 x^{4} + 12 x^{3} + 4 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 41\cdot 1033$ |
$3$ |
$15.1502749166$ |
$823.1937803457944$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$63.3440411985$ |
8.0.2780364800.1 |
$x^{8} - 6 x^{6} - 8 x^{5} + 10 x^{4} + 24 x^{3} + 4 x^{2} + 8 x + 20$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{2}\cdot 1697$ |
$3$ |
$15.1534919536$ |
$481.8156644618649$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$87.4600906248$ |
8.0.2872115200.1 |
$x^{8} - 4 x^{6} - 4 x^{5} + 17 x^{4} - 16 x^{3} + 18 x^{2} - 16 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{2}\cdot 1753$ |
$3$ |
$15.2151147443$ |
$374.4863148367374$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
$[2]$ |
$4$ |
$3$ |
$31.8474482789$ |
8.0.2917728256.2 |
$x^{8} + 6 x^{6} - 12 x^{5} + 13 x^{4} - 24 x^{3} + 44 x^{2} - 36 x + 13$ |
$8$ |
[0,4] |
$2^{16}\cdot 211^{2}$ |
$2$ |
$15.2451115031$ |
$58.1033561853358$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$73.1642624161$ |
8.2.3031367680.1 |
$x^{8} - 4 x^{7} + 6 x^{6} - 2 x^{4} - 20 x^{3} + 2 x^{2} + 8 x - 3$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5\cdot 11\cdot 29^{2}$ |
$4$ |
$15.3180974118$ |
$159.74980438172688$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$4$ |
$79.3357626143$ |
8.0.3436773376.5 |
$x^{8} - 4 x^{7} + 2 x^{6} + 8 x^{5} - 3 x^{4} - 12 x^{3} + 4 x^{2} + 4 x + 5$ |
$8$ |
[0,4] |
$2^{16}\cdot 229^{2}$ |
$2$ |
$15.5603321046$ |
$60.530983801686226$ |
|
|
? |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$85.124441204$ |
8.0.3503882240.6 |
$x^{8} - 21 x^{4} - 16 x^{2} + 185$ |
$8$ |
[0,4] |
$2^{16}\cdot 5\cdot 17^{2}\cdot 37$ |
$4$ |
$15.5979918011$ |
$224.32119828495925$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$4$ |
$3$ |
$52.7367208067$ |
8.0.3572563968.1 |
$x^{8} - 24 x^{6} + 194 x^{4} - 600 x^{2} + 673$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 673$ |
$3$ |
$15.6358862286$ |
$179.73313550928776$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) |
trivial |
$12$ |
$3$ |
$187.714428389$ |
8.4.3582525440.1 |
$x^{8} - 6 x^{6} - 8 x^{5} - 3 x^{4} + 20 x^{3} + 36 x^{2} - 27$ |
$8$ |
[4,2] |
$2^{16}\cdot 5\cdot 13\cdot 29^{2}$ |
$4$ |
$15.6413293351$ |
$173.6663467687393$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$5$ |
$77.1972984677$ |
8.0.3619225600.2 |
$x^{8} + 2 x^{6} + 15 x^{4} - 2 x^{2} + 9$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{2}\cdot 47^{2}$ |
$3$ |
$15.6612692802$ |
$61.318838867023565$ |
|
|
|
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$149.280568435$ |
8.0.3645440000.13 |
$x^{8} - 4 x^{6} + 3 x^{4} - 28 x^{2} + 89$ |
$8$ |
[0,4] |
$2^{16}\cdot 5^{4}\cdot 89$ |
$3$ |
$15.6754040602$ |
$84.38009243891595$ |
|
|
? |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) |
$[2]$ |
$4$ |
$3$ |
$41.7020798514$ |
8.0.3655401472.4 |
$x^{8} - 8 x^{6} + 27 x^{4} - 64 x^{2} + 193$ |
$8$ |
[0,4] |
$2^{16}\cdot 17^{2}\cdot 193$ |
$3$ |
$15.6807519654$ |
$229.12005586591496$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$4$ |
$3$ |
$52.743171598$ |