Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 140\cdot 311 + 264\cdot 311^{2} + 292\cdot 311^{3} + 264\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 75\cdot 311 + 126\cdot 311^{2} + 139\cdot 311^{3} + 262\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 98 + 148\cdot 311 + 189\cdot 311^{2} + 254\cdot 311^{3} + 25\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 100 + 16\cdot 311 + 90\cdot 311^{2} + 77\cdot 311^{3} + 185\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 212 + 294\cdot 311 + 220\cdot 311^{2} + 233\cdot 311^{3} + 125\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 214 + 162\cdot 311 + 121\cdot 311^{2} + 56\cdot 311^{3} + 285\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 298 + 235\cdot 311 + 184\cdot 311^{2} + 171\cdot 311^{3} + 48\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 309 + 170\cdot 311 + 46\cdot 311^{2} + 18\cdot 311^{3} + 46\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,3)(4,5)(6,7)$ |
| $(1,3,5,7,8,6,4,2)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,3)(4,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
