Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 401 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 103 + 75\cdot 401 + 35\cdot 401^{2} + 66\cdot 401^{3} + 64\cdot 401^{4} + 283\cdot 401^{5} + 288\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 105 + 195\cdot 401 + 204\cdot 401^{2} + 197\cdot 401^{3} + 137\cdot 401^{4} + 401^{5} + 191\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 185 + 144\cdot 401 + 51\cdot 401^{2} + 13\cdot 401^{3} + 363\cdot 401^{4} + 214\cdot 401^{5} + 334\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 187 + 264\cdot 401 + 220\cdot 401^{2} + 144\cdot 401^{3} + 35\cdot 401^{4} + 334\cdot 401^{5} + 236\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 214 + 231\cdot 401 + 114\cdot 401^{2} + 18\cdot 401^{3} + 190\cdot 401^{4} + 298\cdot 401^{5} + 400\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 252 + 36\cdot 401 + 183\cdot 401^{2} + 147\cdot 401^{3} + 62\cdot 401^{4} + 296\cdot 401^{5} + 363\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 261 + 24\cdot 401 + 363\cdot 401^{2} + 42\cdot 401^{3} + 239\cdot 401^{4} + 289\cdot 401^{5} + 313\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 299 + 230\cdot 401 + 30\cdot 401^{2} + 172\cdot 401^{3} + 111\cdot 401^{4} + 287\cdot 401^{5} + 276\cdot 401^{6} +O\left(401^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(6,7)$ |
| $(1,6,2,5,4,7,3,8)$ |
| $(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-4$ |
| $2$ | $2$ | $(5,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(2,3)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ |
| $4$ | $8$ | $(1,6,2,5,4,7,3,8)$ | $0$ |
| $4$ | $8$ | $(1,5,3,6,4,8,2,7)$ | $0$ |
| $4$ | $8$ | $(1,6,3,5,4,7,2,8)$ | $0$ |
| $4$ | $8$ | $(1,5,2,6,4,8,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
