Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 26\cdot 97 + 68\cdot 97^{2} + 41\cdot 97^{3} + 95\cdot 97^{4} + 9\cdot 97^{5} + 31\cdot 97^{6} + 40\cdot 97^{7} + 62\cdot 97^{8} + 13\cdot 97^{9} + 58\cdot 97^{10} + 46\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 84\cdot 97 + 68\cdot 97^{2} + 2\cdot 97^{3} + 26\cdot 97^{4} + 57\cdot 97^{5} + 11\cdot 97^{6} + 30\cdot 97^{7} + 88\cdot 97^{8} + 93\cdot 97^{9} + 87\cdot 97^{10} + 50\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 75\cdot 97 + 97^{2} + 16\cdot 97^{3} + 66\cdot 97^{4} + 19\cdot 97^{5} + 29\cdot 97^{6} + 72\cdot 97^{7} + 88\cdot 97^{8} + 80\cdot 97^{9} + 79\cdot 97^{10} + 71\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 91\cdot 97 + 20\cdot 97^{2} + 43\cdot 97^{3} + 96\cdot 97^{4} + 61\cdot 97^{5} + 96\cdot 97^{6} + 80\cdot 97^{7} + 25\cdot 97^{8} + 58\cdot 97^{9} + 64\cdot 97^{10} + 37\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 5\cdot 97 + 76\cdot 97^{2} + 53\cdot 97^{3} + 35\cdot 97^{5} + 16\cdot 97^{7} + 71\cdot 97^{8} + 38\cdot 97^{9} + 32\cdot 97^{10} + 59\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 21\cdot 97 + 95\cdot 97^{2} + 80\cdot 97^{3} + 30\cdot 97^{4} + 77\cdot 97^{5} + 67\cdot 97^{6} + 24\cdot 97^{7} + 8\cdot 97^{8} + 16\cdot 97^{9} + 17\cdot 97^{10} + 25\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 12\cdot 97 + 28\cdot 97^{2} + 94\cdot 97^{3} + 70\cdot 97^{4} + 39\cdot 97^{5} + 85\cdot 97^{6} + 66\cdot 97^{7} + 8\cdot 97^{8} + 3\cdot 97^{9} + 9\cdot 97^{10} + 46\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 92 + 70\cdot 97 + 28\cdot 97^{2} + 55\cdot 97^{3} + 97^{4} + 87\cdot 97^{5} + 65\cdot 97^{6} + 56\cdot 97^{7} + 34\cdot 97^{8} + 83\cdot 97^{9} + 38\cdot 97^{10} + 50\cdot 97^{11} +O\left(97^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
