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 |
19.3.146...972.1 |
$x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} - 19 x^{9} - 20 x^{8} + 32 x^{7} - 9 x^{6} - 17 x^{5} + 17 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 1$ |
$19$ |
[3,8] |
$2^{2}\cdot 139\cdot 126311\cdot 1021043\cdot 2043393285469$ |
$5$ |
$23.8316032998$ |
|
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$737873.586636$ |
19.3.787...824.1 |
$x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} - 19 x^{9} - 20 x^{8} + 33 x^{7} - 9 x^{6} - 18 x^{5} + 17 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 1$ |
$19$ |
[3,8] |
$2^{8}\cdot 317\cdot 25373\cdot 137363\cdot 27854432176313$ |
$5$ |
$29.3922902441$ |
$13978832686367.559$ |
|
|
? |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$6223827.226$ |
19.3.457...029.1 |
$x^{19} - 3 x - 1$ |
$19$ |
[3,8] |
$15451\cdot 1192721\cdot 1892107519\cdot 1311498030721$ |
$4$ |
$46.37957716$ |
$6762456062138603.0$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$278595712.583$ |
19.3.792...893.1 |
$x^{19} - 6 x - 1$ |
$19$ |
[3,8] |
$1811\cdot 26072749799\cdot 167860372602034140737$ |
$3$ |
$60.8360080874$ |
$8.902802417942211e+16$ |
|
|
? |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$9835253933.24$ |
19.3.108...280.1 |
$x^{19} - 4 x - 2$ |
$19$ |
[3,8] |
$2^{18}\cdot 5\cdot 61\cdot 121171\cdot 1116312139673805569479$ |
$5$ |
$61.8392802784$ |
$391677298949786.44$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$7093397402.12$ |
19.3.108...277.1 |
$x^{19} - 4 x - 1$ |
$19$ |
[3,8] |
$277\cdot 1033\cdot 13915748550233\cdot 2716178228054209$ |
$4$ |
$61.8394363535$ |
$1.0399739561894611e+17$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$4641124269.97$ |
19.3.216...536.1 |
$x^{19} - 8 x - 4$ |
$19$ |
[3,8] |
$2^{18}\cdot 11\cdot 53\cdot 421\cdot 2482463\cdot 135422601683162041$ |
$6$ |
$64.1370024239$ |
$553921988966529.94$ |
|
|
? |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$11233192054.3$ |
19.3.202...793.1 |
$x^{19} - 9 x - 4$ |
$19$ |
[3,8] |
$629113\cdot 661588211623\cdot 487141714838860007$ |
$3$ |
$72.1542090665$ |
$4.502833299974699e+17$ |
|
|
? |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$40073649130.0$ |
19.3.749...269.1 |
$x^{19} - 5 x - 3$ |
$19$ |
[3,8] |
$3^{18}\cdot 7\cdot 1193402926633\cdot 231645392486891$ |
$4$ |
$77.295138271$ |
|
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$64364139389.5$ |
19.3.750...024.1 |
$x^{19} - 5 x - 2$ |
$19$ |
[3,8] |
$2^{19}\cdot 7\cdot 4397\cdot 86323\cdot 197123\cdot 2733044013891403$ |
$6$ |
$77.2992926311$ |
$6206242170459233.0$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$232124171257$ |
19.3.750...021.1 |
$x^{19} - 5 x - 1$ |
$19$ |
[3,8] |
$7\cdot 229\cdot 46\!\cdots\!07$ |
$3$ |
$77.2992954426$ |
$8.662985492790676e+17$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$65190521248.6$ |
19.3.596...440.1 |
$x^{19} - 6 x - 4$ |
$19$ |
[3,8] |
$2^{34}\cdot 5\cdot 41\cdot 2819\cdot 847901\cdot 708003258176033$ |
$6$ |
$86.206384516$ |
|
|
|
? |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$418824820501$ |
19.3.239...573.1 |
$x^{19} - 6 x - 3$ |
$19$ |
[3,8] |
$3^{18}\cdot 5526979\cdot 11347703\cdot 986703355583761$ |
$4$ |
$92.7589984567$ |
$704369917783184.2$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$357341058444$ |
19.3.239...328.1 |
$x^{19} - 6 x - 2$ |
$19$ |
[3,8] |
$2^{18}\cdot 13\cdot 653\cdot 10774134287092701706273089133$ |
$4$ |
$92.7591544255$ |
$1.8441904578636164e+16$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$309808088365$ |
19.3.498...989.1 |
$x^{19} - 7 x - 3$ |
$19$ |
[3,8] |
$3^{16}\cdot 13\cdot 89051486359588962351884125193$ |
$3$ |
$96.4007121306$ |
$3591702077425728.5$ |
|
|
|
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$1426601180060$ |
19.3.112...664.1 |
$x^{19} - 7 x - 2$ |
$19$ |
[3,8] |
$2^{18}\cdot 42\!\cdots\!31$ |
$2$ |
$100.604221139$ |
|
|
|
? |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$1443452533040$ |
19.3.440...757.1 |
$x^{19} - 7 x - 5$ |
$19$ |
[3,8] |
$11\cdot 21881\cdot 1229121979\cdot 1490537448908633894245613$ |
$4$ |
$108.122398181$ |
$2.0999012108204483e+19$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$1067772747070$ |
19.3.448...088.1 |
$x^{19} - 7 x - 4$ |
$19$ |
[3,8] |
$2^{18}\cdot 13\cdot 23\cdot 31\cdot 942847\cdot 195714118629896492875189$ |
$6$ |
$108.217286817$ |
$8.271382918944299e+16$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$5512434609740$ |
19.3.448...653.1 |
$x^{19} - 7 x - 1$ |
$19$ |
[3,8] |
$13\cdot 34\!\cdots\!81$ |
$2$ |
$108.21901362$ |
$2.1177950362847265e+19$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$982033676331$ |
19.3.546...464.1 |
$x^{19} - 8 x - 6$ |
$19$ |
[3,8] |
$2^{18}\cdot 3^{18}\cdot 1626601603931\cdot 33108756840559$ |
$4$ |
$123.444252011$ |
|
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$6352477156150$ |
19.3.566...853.1 |
$x^{19} - 8 x - 5$ |
$19$ |
[3,8] |
$10771\cdot 2100606749\cdot 14133445561\cdot 17708737489650787$ |
$4$ |
$123.670203498$ |
$7.52520293934325e+19$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$7376942071750$ |
19.3.567...997.1 |
$x^{19} - 8 x - 3$ |
$19$ |
[3,8] |
$3^{18}\cdot 4967\cdot 151598782807\cdot 19437585947910517$ |
$4$ |
$123.678871828$ |
|
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$7514560123740$ |
19.3.567...752.1 |
$x^{19} - 8 x - 2$ |
$19$ |
[3,8] |
$2^{18}\cdot 269879\cdot 934176914887\cdot 85797908177168821$ |
$4$ |
$123.678872708$ |
$2.836114583447896e+17$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$11927980539300$ |
19.3.567...749.1 |
$x^{19} - 8 x - 1$ |
$19$ |
[3,8] |
$29\cdot 6080707\cdot 32\!\cdots\!83$ |
$3$ |
$123.678872708$ |
$7.53021580028138e+19$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$5830581685330$ |
19.3.499...165.1 |
$x^{19} - 9 x - 7$ |
$19$ |
[3,8] |
$5\cdot 99\!\cdots\!33$ |
$2$ |
$138.68158308$ |
$2.2344904891971738e+20$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$12522074027400$ |
19.3.529...472.1 |
$x^{19} - 9 x - 6$ |
$19$ |
[3,8] |
$2^{19}\cdot 3^{18}\cdot 107\cdot 122029\cdot 19964965175341736357$ |
$5$ |
$139.110998501$ |
|
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$55933455099100$ |
19.3.531...861.1 |
$x^{19} - 9 x - 5$ |
$19$ |
[3,8] |
$6219297888079\cdot 8544954240347693873940241859$ |
$2$ |
$139.137691902$ |
$2.305289913670862e+20$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$27546004978300$ |
19.3.531...005.1 |
$x^{19} - 9 x - 3$ |
$19$ |
[3,8] |
$3^{18}\cdot 5\cdot 71\cdot 1069\cdot 2542879\cdot 142166878031666904029$ |
$6$ |
$139.138731691$ |
$3.3164595121250068e+16$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$14455471769800$ |
19.3.531...760.1 |
$x^{19} - 9 x - 2$ |
$19$ |
[3,8] |
$2^{19}\cdot 5\cdot 17\cdot 11\!\cdots\!87$ |
$4$ |
$139.138731797$ |
|
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$67811808493000$ |
19.3.531...757.1 |
$x^{19} - 9 x - 1$ |
$19$ |
[3,8] |
$101\cdot 9221\cdot 1141909\cdot 263051581967\cdot 189994470465354439$ |
$5$ |
$139.138731797$ |
$2.3054535981619066e+20$ |
|
|
✓ |
$S_{19}$ (as 19T8) |
trivial |
$2$ |
$10$ |
$11114856383000$ |
19.3.361...000.1 |
$x^{19} - 380 x - 360$ |
$19$ |
[3,8] |
$2^{24}\cdot 3^{36}\cdot 5^{18}\cdot 19^{20}$ |
$4$ |
$1961.129021$ |
|
|
|
? |
$A_{19}$ (as 19T7) |
trivial |
$2$ |
$10$ |
$3794739065090000000000000$ |