Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 12\cdot 193 + 24\cdot 193^{2} + 118\cdot 193^{3} + 51\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 151\cdot 193 + 188\cdot 193^{2} + 147\cdot 193^{3} + 55\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 + 35\cdot 193 + 74\cdot 193^{2} + 138\cdot 193^{3} + 165\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 90 + 91\cdot 193 + 119\cdot 193^{2} + 122\cdot 193^{3} + 133\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 97 + 169\cdot 193 + 124\cdot 193^{2} + 167\cdot 193^{3} + 86\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 134 + 50\cdot 193 + 127\cdot 193^{2} + 104\cdot 193^{3} + 69\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 171 + 107\cdot 193 + 161\cdot 193^{2} + 106\cdot 193^{3} + 159\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 190 + 153\cdot 193 + 144\cdot 193^{2} + 58\cdot 193^{3} + 49\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(4,6)$ |
| $(1,5)(2,7)(3,8)(4,6)$ |
| $(1,3,6,7,5,8,4,2)$ |
| $(1,6,5,4)(2,3,7,8)$ |
| $(1,4)(2,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-4$ |
| $2$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,5,4)(2,3,7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,5,6)(2,3,7,8)$ | $0$ |
| $4$ | $4$ | $(1,3,5,8)(2,4,7,6)$ | $0$ |
| $4$ | $8$ | $(1,3,6,7,5,8,4,2)$ | $0$ |
| $4$ | $8$ | $(1,3,4,2,5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
