Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 691 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 387 + 637\cdot 691 + 635\cdot 691^{2} + 584\cdot 691^{3} +O\left(691^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 428 + 44\cdot 691 + 93\cdot 691^{2} + 400\cdot 691^{3} + 608\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 481 + 117\cdot 691 + 457\cdot 691^{2} + 249\cdot 691^{3} + 460\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 485 + 298\cdot 691 + 73\cdot 691^{2} + 364\cdot 691^{3} + 212\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 526 + 396\cdot 691 + 221\cdot 691^{2} + 179\cdot 691^{3} + 129\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 577 + 607\cdot 691 + 667\cdot 691^{2} + 322\cdot 691^{3} + 327\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 584 + 430\cdot 691 + 547\cdot 691^{2} + 294\cdot 691^{3} + 233\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 680 + 229\cdot 691 + 67\cdot 691^{2} + 368\cdot 691^{3} + 100\cdot 691^{4} +O\left(691^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(2,4)(6,7)$ |
| $(1,6,2,3,5,7,4,8)$ |
| $(3,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,8)(6,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,5)(2,4)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,4)(3,7,8,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,4)(3,6,8,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,2,3,5,7,4,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,4,6,5,8,2,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,2,6,5,8,4,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,4,3,5,7,2,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
