Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 421 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 63 + 193\cdot 421 + 18\cdot 421^{2} + 46\cdot 421^{3} + 182\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 82 + 199\cdot 421 + 122\cdot 421^{2} + 316\cdot 421^{3} + 357\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 259 + 52\cdot 421 + 18\cdot 421^{2} + 13\cdot 421^{3} + 103\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 263 + 258\cdot 421 + 81\cdot 421^{2} + 49\cdot 421^{3} + 96\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 274 + 22\cdot 421 + 300\cdot 421^{2} + 190\cdot 421^{3} + 5\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 344 + 299\cdot 421 + 372\cdot 421^{2} + 198\cdot 421^{3} + 355\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 408 + 369\cdot 421 + 363\cdot 421^{2} + 257\cdot 421^{3} + 156\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 413 + 287\cdot 421 + 406\cdot 421^{2} + 190\cdot 421^{3} + 6\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,4)(3,5)(6,8)$ |
| $(1,2,5,6)(3,8,7,4)$ |
| $(3,7)(4,8)$ |
| $(1,7,6,4,5,3,2,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $2$ | $2$ | $(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(2,6)(3,4)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,6,5,2)(3,8,7,4)$ | $0$ |
| $2$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | $0$ |
| $4$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $0$ |
| $4$ | $8$ | $(1,7,6,4,5,3,2,8)$ | $0$ |
| $4$ | $8$ | $(1,3,2,4,5,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
