Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 409 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 96 + 186\cdot 409 + 176\cdot 409^{2} + 11\cdot 409^{3} + 155\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 150 + 323\cdot 409 + 37\cdot 409^{2} + 151\cdot 409^{3} + 99\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 236 + 183\cdot 409 + 294\cdot 409^{2} + 212\cdot 409^{3} + 364\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 257 + 203\cdot 409 + 187\cdot 409^{2} + 223\cdot 409^{3} + 258\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 281 + 62\cdot 409 + 235\cdot 409^{2} + 48\cdot 409^{3} + 257\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 286 + 54\cdot 409 + 100\cdot 409^{2} + 325\cdot 409^{3} + 276\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 337 + 124\cdot 409 + 309\cdot 409^{2} + 33\cdot 409^{3} + 199\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 404 + 87\cdot 409 + 295\cdot 409^{2} + 220\cdot 409^{3} + 25\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,8)(5,6)$ |
| $(1,7)(4,5)(6,8)$ |
| $(1,5,7,6)(2,4,3,8)$ |
| $(1,7)(2,3)(4,8)(5,6)$ |
| $(1,2,7,3)(4,5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,3)(4,8)(5,6)$ |
$-4$ |
| $2$ |
$2$ |
$(4,8)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(4,5)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,4)(3,8)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(4,6)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,7,3)(4,5,8,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,7,3)(4,6,8,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,7,6)(2,4,3,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,3,6,7,8,2,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,2,5,7,8,3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
