Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 74 + 9\cdot 277 + 18\cdot 277^{2} + 222\cdot 277^{3} + 192\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 76 + 156\cdot 277 + 30\cdot 277^{2} + 84\cdot 277^{3} + 74\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 101 + 254\cdot 277 + 11\cdot 277^{2} + 195\cdot 277^{3} + 86\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 120 + 107\cdot 277 + 151\cdot 277^{2} + 69\cdot 277^{3} + 74\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 227 + 186\cdot 277 + 33\cdot 277^{2} + 42\cdot 277^{3} + 143\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 258 + 35\cdot 277 + 83\cdot 277^{2} + 205\cdot 277^{3} + 41\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 260 + 174\cdot 277 + 172\cdot 277^{2} + 49\cdot 277^{3} + 175\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 271 + 182\cdot 277 + 52\cdot 277^{2} + 240\cdot 277^{3} + 42\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(4,6)$ |
| $(1,8)(2,4)(3,6)(5,7)$ |
| $(1,2,7,3)(4,5,6,8)$ |
| $(1,7)(2,3)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,3)(4,6)(5,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(4,6)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,8)(5,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,4,7,6)(2,5,3,8)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,6,7,4)(2,8,3,5)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,7,3)(4,5,6,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,7,5)(2,6,3,4)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,6,7,4)(2,5,3,8)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
