Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 509 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 112 + 36\cdot 509 + 215\cdot 509^{2} + 422\cdot 509^{3} + 67\cdot 509^{4} + 213\cdot 509^{5} + 314\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 119 + 26\cdot 509 + 469\cdot 509^{2} + 197\cdot 509^{3} + 54\cdot 509^{4} + 497\cdot 509^{5} + 465\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 183 + 295\cdot 509 + 444\cdot 509^{2} + 421\cdot 509^{3} + 18\cdot 509^{4} + 172\cdot 509^{5} + 353\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 190 + 285\cdot 509 + 189\cdot 509^{2} + 197\cdot 509^{3} + 5\cdot 509^{4} + 456\cdot 509^{5} + 504\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 249 + 109\cdot 509 + 25\cdot 509^{2} + 272\cdot 509^{3} + 482\cdot 509^{4} + 102\cdot 509^{5} + 61\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 334 + 74\cdot 509 + 407\cdot 509^{2} + 140\cdot 509^{3} + 81\cdot 509^{4} + 115\cdot 509^{5} + 497\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 384 + 112\cdot 509 + 206\cdot 509^{2} + 257\cdot 509^{3} + 354\cdot 509^{4} + 233\cdot 509^{5} + 210\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 469 + 77\cdot 509 + 79\cdot 509^{2} + 126\cdot 509^{3} + 462\cdot 509^{4} + 245\cdot 509^{5} + 137\cdot 509^{6} +O\left(509^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,7,4,6)(2,8,3,5)$ |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
| $(1,2,4,3)(5,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,3)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,2,4,3)(5,7,8,6)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,3,4,2)(5,6,8,7)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,4,3)(5,6,8,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,2,4,3)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,2,4,3)(5,8)(6,7)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,3,4,2)(5,8)(6,7)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $4$ |
$4$ |
$(1,7,4,6)(2,8,3,5)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,7,2,8,4,6,3,5)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,8,3,7,4,5,2,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
