Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 101 + 196\cdot 389 + 261\cdot 389^{2} + 326\cdot 389^{3} + 135\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 112 + 105\cdot 389 + 46\cdot 389^{2} + 130\cdot 389^{3} + 327\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 167 + 284\cdot 389 + 69\cdot 389^{2} + 104\cdot 389^{3} + 96\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 217 + 386\cdot 389 + 14\cdot 389^{2} + 293\cdot 389^{3} + 275\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 233 + 111\cdot 389 + 174\cdot 389^{2} + 372\cdot 389^{3} + 257\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 355 + 375\cdot 389 + 365\cdot 389^{2} + 103\cdot 389^{3} + 160\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 381 + 197\cdot 389 + 264\cdot 389^{2} + 286\cdot 389^{3} + 151\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 383 + 286\cdot 389 + 358\cdot 389^{2} + 327\cdot 389^{3} + 150\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,8)(3,5)(4,7)$ |
| $(1,4,6,7)(2,3,8,5)$ |
| $(2,8)(3,5)$ |
| $(1,8,7,5,6,2,4,3)$ |
| $(1,2,6,8)(3,7,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $-4$ |
| $2$ | $2$ | $(2,8)(3,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(3,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,6,4)(2,3,8,5)$ | $0$ |
| $2$ | $4$ | $(1,4,6,7)(2,3,8,5)$ | $0$ |
| $4$ | $4$ | $(1,8,6,2)(3,4,5,7)$ | $0$ |
| $4$ | $8$ | $(1,8,7,5,6,2,4,3)$ | $0$ |
| $4$ | $8$ | $(1,5,7,2,6,3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
