Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 5\cdot 7 + 3\cdot 7^{2} + 2\cdot 7^{3} + 4\cdot 7^{4} + 2\cdot 7^{5} + 6\cdot 7^{6} + 6\cdot 7^{7} +O\left(7^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 6\cdot 7 + 5\cdot 7^{2} + 4\cdot 7^{3} + 3\cdot 7^{4} + 6\cdot 7^{5} + 2\cdot 7^{6} + 7^{7} +O\left(7^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 7^{2} + 2\cdot 7^{3} + 3\cdot 7^{4} + 4\cdot 7^{6} + 5\cdot 7^{7} +O\left(7^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 7 + 3\cdot 7^{2} + 4\cdot 7^{3} + 2\cdot 7^{4} + 4\cdot 7^{5} +O\left(7^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
