Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 188\cdot 379 + 177\cdot 379^{2} + 174\cdot 379^{3} + 171\cdot 379^{4} + 276\cdot 379^{5} + 296\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 154 + 80\cdot 379 + 48\cdot 379^{2} + 155\cdot 379^{3} + 168\cdot 379^{4} + 296\cdot 379^{5} + 284\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 221 + 122\cdot 379 + 71\cdot 379^{2} + 172\cdot 379^{3} + 138\cdot 379^{4} + 191\cdot 379^{5} + 357\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 249 + 219\cdot 379 + 233\cdot 379^{2} + 192\cdot 379^{3} + 326\cdot 379^{4} + 19\cdot 379^{5} + 327\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 252 + 157\cdot 379 + 128\cdot 379^{2} + 270\cdot 379^{3} + 355\cdot 379^{4} + 321\cdot 379^{5} + 112\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 266 + 90\cdot 379 + 188\cdot 379^{2} + 35\cdot 379^{3} + 174\cdot 379^{4} + 124\cdot 379^{5} + 305\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 339 + 366\cdot 379 + 81\cdot 379^{2} + 256\cdot 379^{3} + 279\cdot 379^{4} + 372\cdot 379^{5} + 197\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 371 + 289\cdot 379 + 207\cdot 379^{2} + 259\cdot 379^{3} + 280\cdot 379^{4} + 291\cdot 379^{5} + 12\cdot 379^{6} +O\left(379^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,7,4)(2,8,3,6)$ |
| $(1,7)(6,8)$ |
| $(2,3)(4,5)$ |
| $(4,5)(6,8)$ |
| $(1,6)(2,4)(3,5)(7,8)$ |
| $(4,8,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,3)(4,5)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(4,5)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,4)(3,5)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,7)(4,6)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,5)(3,4)(7,8)$ |
$0$ |
| $8$ |
$2$ |
$(1,7)(4,6)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,2)(4,8,5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,2)(4,6,5,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,7,4)(2,8,3,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,7,8)(2,4,3,5)$ |
$0$ |
| $4$ |
$4$ |
$(4,8,5,6)$ |
$2$ |
| $4$ |
$4$ |
$(1,7)(2,3)(4,6,5,8)$ |
$-2$ |
| $8$ |
$8$ |
$(1,8,3,5,7,6,2,4)$ |
$0$ |
| $8$ |
$8$ |
$(1,8,3,4,7,6,2,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
