Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 159\cdot 229 + 219\cdot 229^{2} + 127\cdot 229^{3} + 128\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 + 109\cdot 229 + 169\cdot 229^{2} + 27\cdot 229^{3} + 184\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 108 + 215\cdot 229 + 225\cdot 229^{2} + 165\cdot 229^{3} + 220\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 116 + 122\cdot 229 + 229^{2} + 110\cdot 229^{3} + 138\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 191 + 42\cdot 229 + 113\cdot 229^{2} + 124\cdot 229^{3} + 134\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 212 + 8\cdot 229 + 6\cdot 229^{2} + 84\cdot 229^{3} + 140\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 213 + 12\cdot 229 + 118\cdot 229^{2} + 142\cdot 229^{3} + 72\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 215 + 15\cdot 229 + 62\cdot 229^{2} + 133\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 |
| $(1,4)(2,3)(5,7)(6,8)$ |
| $(1,7)(2,8)(3,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,2,7,8)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,7,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,7,3)(2,4,8,5)$ | $0$ |
| $2$ | $4$ | $(1,2,7,8)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,5,7,4)(2,6,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
