Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 1033 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 226 + 738\cdot 1033 + 5\cdot 1033^{2} + 286\cdot 1033^{3} + 40\cdot 1033^{4} + 799\cdot 1033^{5} + 10\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 247 + 217\cdot 1033 + 620\cdot 1033^{2} + 372\cdot 1033^{3} + 10\cdot 1033^{4} + 182\cdot 1033^{5} + 611\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 333 + 102\cdot 1033 + 627\cdot 1033^{2} + 454\cdot 1033^{3} + 907\cdot 1033^{4} + 49\cdot 1033^{5} + 617\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 391 + 261\cdot 1033 + 924\cdot 1033^{2} + 984\cdot 1033^{3} + 340\cdot 1033^{4} + 435\cdot 1033^{5} + 864\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 624 + 173\cdot 1033 + 526\cdot 1033^{2} + 752\cdot 1033^{3} + 840\cdot 1033^{4} + 734\cdot 1033^{5} + 913\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 640 + 402\cdot 1033 + 168\cdot 1033^{2} + 862\cdot 1033^{3} + 664\cdot 1033^{4} + 249\cdot 1033^{5} + 360\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 751 + 203\cdot 1033 + 308\cdot 1033^{2} + 868\cdot 1033^{3} + 340\cdot 1033^{4} + 256\cdot 1033^{5} + 348\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 923 + 999\cdot 1033 + 951\cdot 1033^{2} + 583\cdot 1033^{3} + 986\cdot 1033^{4} + 391\cdot 1033^{5} + 406\cdot 1033^{6} +O\left(1033^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(6,7)$ |
| $(1,4)(5,8)$ |
| $(1,5)(4,8)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
| $(1,4)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-4$ |
| $2$ |
$2$ |
$(2,3)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,4)(5,6)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(2,7)(3,6)$ |
$-2$ |
| $4$ |
$2$ |
$(1,4)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$2$ |
| $4$ |
$4$ |
$(1,3,4,2)(5,6,8,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,4,3)(5,6,8,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,4,8)(2,7,3,6)$ |
$0$ |
| $8$ |
$4$ |
$(1,3,5,6)(2,8,7,4)$ |
$0$ |
| $8$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
| $8$ |
$4$ |
$(1,4)(2,7,3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
