Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 79\cdot 277 + 249\cdot 277^{2} + 165\cdot 277^{3} + 264\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 + 185\cdot 277 + 23\cdot 277^{2} + 15\cdot 277^{3} + 79\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 104 + 156\cdot 277 + 266\cdot 277^{2} + 2\cdot 277^{3} + 214\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 127 + 14\cdot 277 + 236\cdot 277^{2} + 147\cdot 277^{3} + 248\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 151 + 262\cdot 277 + 40\cdot 277^{2} + 129\cdot 277^{3} + 28\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 174 + 120\cdot 277 + 10\cdot 277^{2} + 274\cdot 277^{3} + 62\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 214 + 91\cdot 277 + 253\cdot 277^{2} + 261\cdot 277^{3} + 197\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 261 + 197\cdot 277 + 27\cdot 277^{2} + 111\cdot 277^{3} + 12\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
