Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 51 + 190\cdot 379 + 162\cdot 379^{2} + 109\cdot 379^{3} + 73\cdot 379^{4} + 298\cdot 379^{5} + 225\cdot 379^{6} + 84\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 91 + 267\cdot 379 + 146\cdot 379^{2} + 148\cdot 379^{3} + 240\cdot 379^{4} + 36\cdot 379^{5} + 192\cdot 379^{6} + 17\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 96 + 137\cdot 379 + 119\cdot 379^{2} + 182\cdot 379^{3} + 218\cdot 379^{4} + 54\cdot 379^{5} + 289\cdot 379^{6} + 309\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 97 + 41\cdot 379 + 325\cdot 379^{2} + 77\cdot 379^{3} + 328\cdot 379^{4} + 111\cdot 379^{5} + 243\cdot 379^{6} + 272\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 121 + 191\cdot 379 + 164\cdot 379^{2} + 46\cdot 379^{3} + 226\cdot 379^{4} + 89\cdot 379^{5} + 42\cdot 379^{6} + 202\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 123 + 354\cdot 379 + 224\cdot 379^{2} + 239\cdot 379^{3} + 54\cdot 379^{4} + 376\cdot 379^{5} + 314\cdot 379^{6} + 165\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 271 + 23\cdot 379 + 178\cdot 379^{2} + 156\cdot 379^{3} + 53\cdot 379^{4} + 97\cdot 379^{5} + 260\cdot 379^{6} + 242\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 288 + 310\cdot 379 + 194\cdot 379^{2} + 176\cdot 379^{3} + 321\cdot 379^{4} + 72\cdot 379^{5} + 327\cdot 379^{6} + 220\cdot 379^{7} +O\left(379^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,4,2)(3,7,8,6)$ |
| $(1,4)(3,8)$ |
| $(1,6,2,3,4,7,5,8)$ |
| $(2,5)(6,7)$ |
| $(2,5)(3,8)$ |
| $(1,6)(2,8)(3,5)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-4$ |
| $2$ | $2$ | $(1,4)(2,5)$ | $0$ |
| $4$ | $2$ | $(2,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $0$ |
| $8$ | $2$ | $(1,5)(2,4)(3,8)$ | $0$ |
| $2$ | $4$ | $(1,5,4,2)(3,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,5)(3,7,8,6)$ | $0$ |
| $4$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $0$ |
| $4$ | $4$ | $(1,7,4,6)(2,8,5,3)$ | $0$ |
| $4$ | $4$ | $(1,2,4,5)(3,8)(6,7)$ | $-2$ |
| $4$ | $4$ | $(1,2,4,5)$ | $2$ |
| $8$ | $8$ | $(1,6,2,3,4,7,5,8)$ | $0$ |
| $8$ | $8$ | $(1,7,5,8,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
