Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 331 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 + 182\cdot 331 + 18\cdot 331^{2} + 313\cdot 331^{3} + 270\cdot 331^{4} + 231\cdot 331^{5} + 92\cdot 331^{6} + 75\cdot 331^{7} + 221\cdot 331^{8} + 12\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 + 255\cdot 331 + 42\cdot 331^{2} + 198\cdot 331^{3} + 253\cdot 331^{4} + 243\cdot 331^{5} + 36\cdot 331^{6} + 296\cdot 331^{7} + 22\cdot 331^{8} + 44\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 119 + 111\cdot 331 + 146\cdot 331^{2} + 219\cdot 331^{3} + 272\cdot 331^{4} + 162\cdot 331^{5} + 326\cdot 331^{6} + 163\cdot 331^{7} + 197\cdot 331^{8} + 299\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 143 + 88\cdot 331 + 152\cdot 331^{2} + 155\cdot 331^{3} + 165\cdot 331^{4} + 218\cdot 331^{5} + 260\cdot 331^{6} + 244\cdot 331^{7} + 329\cdot 331^{8} + 329\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 188 + 242\cdot 331 + 178\cdot 331^{2} + 175\cdot 331^{3} + 165\cdot 331^{4} + 112\cdot 331^{5} + 70\cdot 331^{6} + 86\cdot 331^{7} + 331^{8} + 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 212 + 219\cdot 331 + 184\cdot 331^{2} + 111\cdot 331^{3} + 58\cdot 331^{4} + 168\cdot 331^{5} + 4\cdot 331^{6} + 167\cdot 331^{7} + 133\cdot 331^{8} + 31\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 253 + 75\cdot 331 + 288\cdot 331^{2} + 132\cdot 331^{3} + 77\cdot 331^{4} + 87\cdot 331^{5} + 294\cdot 331^{6} + 34\cdot 331^{7} + 308\cdot 331^{8} + 286\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 305 + 148\cdot 331 + 312\cdot 331^{2} + 17\cdot 331^{3} + 60\cdot 331^{4} + 99\cdot 331^{5} + 238\cdot 331^{6} + 255\cdot 331^{7} + 109\cdot 331^{8} + 318\cdot 331^{9} +O\left(331^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(1,2,3,5,8,7,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,2,3,5,8,7,6,4)$ | $0$ |
| $4$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $0$ |
| $4$ | $8$ | $(1,2,3,4,8,7,6,5)$ | $0$ |
| $4$ | $8$ | $(1,4,6,2,8,5,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
