Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 461 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 243\cdot 461 + 158\cdot 461^{2} + 350\cdot 461^{3} + 70\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 + 113\cdot 461 + 238\cdot 461^{2} + 184\cdot 461^{3} + 361\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 52 + 55\cdot 461 + 398\cdot 461^{2} + 421\cdot 461^{3} + 433\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 133 + 125\cdot 461 + 117\cdot 461^{2} + 12\cdot 461^{3} + 168\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 162 + 240\cdot 461 + 30\cdot 461^{2} + 184\cdot 461^{3} + 390\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 217 + 423\cdot 461 + 200\cdot 461^{2} + 439\cdot 461^{3} + 380\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 380 + 139\cdot 461 + 236\cdot 461^{3} + 266\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 387 + 42\cdot 461 + 239\cdot 461^{2} + 15\cdot 461^{3} + 233\cdot 461^{4} +O\left(461^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,4)(7,8)$ |
| $(1,8,3,5,6,7,4,2)$ |
| $(2,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-4$ |
| $2$ | $2$ | $(2,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(3,4)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,8,5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,7,5,8)$ | $0$ |
| $4$ | $8$ | $(1,8,3,5,6,7,4,2)$ | $0$ |
| $4$ | $8$ | $(1,5,4,8,6,2,3,7)$ | $0$ |
| $4$ | $8$ | $(1,8,4,5,6,7,3,2)$ | $0$ |
| $4$ | $8$ | $(1,5,3,8,6,2,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
