Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 409 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 81 + 183\cdot 409 + 53\cdot 409^{2} + 136\cdot 409^{3} + 203\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 91 + 238\cdot 409 + 53\cdot 409^{2} + 214\cdot 409^{3} + 373\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 181 + 184\cdot 409 + 129\cdot 409^{2} + 321\cdot 409^{3} + 380\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 274 + 314\cdot 409 + 111\cdot 409^{2} + 311\cdot 409^{3} + 103\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 311 + 309\cdot 409 + 275\cdot 409^{2} + 169\cdot 409^{3} + 48\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 323 + 44\cdot 409 + 193\cdot 409^{2} + 335\cdot 409^{3} + 361\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 388 + 34\cdot 409 + 74\cdot 409^{2} + 334\cdot 409^{3} + 403\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 396 + 325\cdot 409 + 335\cdot 409^{2} + 222\cdot 409^{3} + 169\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,3,6,5,2,7,8)$ |
| $(1,5)(2,4)(3,7)(6,8)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(2,4)(6,8)$ |
| $(1,7,5,3)(2,8,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-4$ |
| $2$ | $2$ | $(2,4)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(3,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,8,4,6)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,6,4,8)$ | $0$ |
| $4$ | $4$ | $(1,2,5,4)(3,6,7,8)$ | $0$ |
| $4$ | $8$ | $(1,8,7,2,5,6,3,4)$ | $0$ |
| $4$ | $8$ | $(1,8,3,4,5,6,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
