Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 209\cdot 229 + 166\cdot 229^{2} + 191\cdot 229^{3} + 17\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 118\cdot 229 + 33\cdot 229^{2} + 215\cdot 229^{3} + 49\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 150\cdot 229 + 10\cdot 229^{2} + 192\cdot 229^{3} + 148\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 + 97\cdot 229 + 120\cdot 229^{2} + 180\cdot 229^{3} + 144\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 + 206\cdot 229 + 217\cdot 229^{2} + 32\cdot 229^{3} + 176\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 137 + 33\cdot 229 + 137\cdot 229^{2} + 99\cdot 229^{3} + 16\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 158 + 137\cdot 229 + 6\cdot 229^{2} + 15\cdot 229^{3} + 7\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 170 + 192\cdot 229 + 222\cdot 229^{2} + 217\cdot 229^{3} + 125\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,4)(5,8)$ |
| $(1,3,6,7)(2,8,4,5)$ |
| $(1,6)(2,4)(3,7)(5,8)$ |
| $(1,2,6,4)(3,8,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(2,4)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,3,6,7)(2,8,4,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,6,3)(2,5,4,8)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,6,4)(3,8,7,5)$ | $0$ |
| $2$ | $4$ | $(1,3,6,7)(2,5,4,8)$ | $0$ |
| $2$ | $4$ | $(1,8,6,5)(2,3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
