Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 751 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 196\cdot 751 + 731\cdot 751^{2} + 718\cdot 751^{3} + 530\cdot 751^{4} + 33\cdot 751^{5} + 495\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 129 + 18\cdot 751 + 363\cdot 751^{2} + 103\cdot 751^{3} + 63\cdot 751^{4} + 477\cdot 751^{5} + 741\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 203 + 411\cdot 751 + 393\cdot 751^{2} + 421\cdot 751^{3} + 62\cdot 751^{4} + 338\cdot 751^{5} + 130\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 228 + 217\cdot 751 + 685\cdot 751^{2} + 15\cdot 751^{3} + 427\cdot 751^{4} + 585\cdot 751^{5} + 699\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 523 + 533\cdot 751 + 65\cdot 751^{2} + 735\cdot 751^{3} + 323\cdot 751^{4} + 165\cdot 751^{5} + 51\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 548 + 339\cdot 751 + 357\cdot 751^{2} + 329\cdot 751^{3} + 688\cdot 751^{4} + 412\cdot 751^{5} + 620\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 622 + 732\cdot 751 + 387\cdot 751^{2} + 647\cdot 751^{3} + 687\cdot 751^{4} + 273\cdot 751^{5} + 9\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 698 + 554\cdot 751 + 19\cdot 751^{2} + 32\cdot 751^{3} + 220\cdot 751^{4} + 717\cdot 751^{5} + 255\cdot 751^{6} +O\left(751^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,5,6,4)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,5)(2,3)(4,8)(6,7)$ |
| $(3,6)(4,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(3,5,6,4)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(3,4,6,5)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,6)(4,5)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,6)(4,5)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $4$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,6,7,4,8,3,2,5)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,4,2,6,8,5,7,3)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
