Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 521 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 56 + 142\cdot 521 + 414\cdot 521^{2} + 159\cdot 521^{3} + 94\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 135 + 433\cdot 521 + 189\cdot 521^{2} + 75\cdot 521^{3} + 260\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 176 + 494\cdot 521 + 178\cdot 521^{2} + 111\cdot 521^{3} + 453\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 201 + 219\cdot 521 + 200\cdot 521^{2} + 334\cdot 521^{3} + 160\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 320 + 301\cdot 521 + 320\cdot 521^{2} + 186\cdot 521^{3} + 360\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 345 + 26\cdot 521 + 342\cdot 521^{2} + 409\cdot 521^{3} + 67\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 386 + 87\cdot 521 + 331\cdot 521^{2} + 445\cdot 521^{3} + 260\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 465 + 378\cdot 521 + 106\cdot 521^{2} + 361\cdot 521^{3} + 426\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,4,7,6,8,5,2,3)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(2,7)(3,5)(4,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,7,6,8,5,2,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,7,5,8,3,2,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
