Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 59\cdot 379 + 135\cdot 379^{2} + 344\cdot 379^{3} + 86\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 145\cdot 379 + 286\cdot 379^{2} + 309\cdot 379^{3} + 64\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 102 + 254\cdot 379 + 77\cdot 379^{2} + 46\cdot 379^{3} + 9\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 237 + 12\cdot 379 + 302\cdot 379^{2} + 247\cdot 379^{3} + 79\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 254 + 367\cdot 379 + 217\cdot 379^{2} + 54\cdot 379^{3} + 108\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 265 + 177\cdot 379 + 173\cdot 379^{2} + 250\cdot 379^{3} + 321\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 291 + 346\cdot 379 + 244\cdot 379^{2} + 166\cdot 379^{3} + 343\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 325 + 152\cdot 379 + 78\cdot 379^{2} + 96\cdot 379^{3} + 123\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,4)(3,8)(5,7)$ |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(2,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(2,5)(6,8)$ |
$0$ |
$0$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,8)(6,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,4,3,7)(2,8,5,6)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,7,3,4)(2,6,5,8)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,8,7,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,3,6)(2,4,5,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,7)(2,6,5,8)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
