Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 + 58\cdot 311 + 147\cdot 311^{2} + 199\cdot 311^{3} + 12\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 215\cdot 311 + 199\cdot 311^{2} + 159\cdot 311^{3} + 9\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 117 + 310\cdot 311 + 110\cdot 311^{2} + 137\cdot 311^{3} + 246\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 161 + 33\cdot 311 + 154\cdot 311^{2} + 180\cdot 311^{3} + 156\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 170 + 290\cdot 311 + 105\cdot 311^{2} + 43\cdot 311^{3} + 186\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 181 + 225\cdot 311 + 278\cdot 311^{2} + 200\cdot 311^{3} + 183\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 256 + 41\cdot 311 + 79\cdot 311^{2} + 142\cdot 311^{3} + 228\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 264 + 68\cdot 311 + 168\cdot 311^{2} + 180\cdot 311^{3} + 220\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,7,3,4)(2,6,8,5)$ |
| $(1,2,4,5,3,8,7,6)$ |
| $(1,5)(2,4)(3,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(3,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,3,4)(2,6,8,5)$ | $0$ |
| $2$ | $8$ | $(1,6,7,8,3,5,4,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,4,6,3,2,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
