Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 50\cdot 97 + 40\cdot 97^{2} + 7\cdot 97^{3} + 37\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 33\cdot 97 + 9\cdot 97^{2} + 39\cdot 97^{3} + 56\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 50\cdot 97 + 29\cdot 97^{2} + 76\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 53\cdot 97 + 40\cdot 97^{2} + 2\cdot 97^{3} + 90\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 57\cdot 97 + 17\cdot 97^{2} + 76\cdot 97^{3} + 33\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 + 90\cdot 97 + 68\cdot 97^{2} + 27\cdot 97^{3} + 32\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 62 + 47\cdot 97 + 96\cdot 97^{2} + 30\cdot 97^{3} + 22\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 78 + 5\cdot 97 + 85\cdot 97^{2} + 30\cdot 97^{3} + 5\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6,8)(2,4,5,3)$ |
| $(1,8)(2,5)(6,7)$ |
| $(1,5,6,2)(3,7,4,8)$ |
| $(1,6)(2,5)(3,4)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,7,6,8)(2,4,5,3)$ | $0$ |
| $4$ | $4$ | $(1,5,6,2)(3,7,4,8)$ | $0$ |
| $2$ | $8$ | $(1,2,8,3,6,5,7,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,8,4,6,2,7,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
