Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 3\cdot 7 + 2\cdot 7^{2} + 5\cdot 7^{3} + 4\cdot 7^{4} + 5\cdot 7^{5} + 5\cdot 7^{6} + 6\cdot 7^{7} +O\left(7^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 3\cdot 7 + 4\cdot 7^{2} + 7^{3} + 3\cdot 7^{4} + 5\cdot 7^{5} + 4\cdot 7^{6} +O\left(7^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 3\cdot 7 + 2\cdot 7^{2} + 5\cdot 7^{3} + 3\cdot 7^{4} + 7^{5} + 2\cdot 7^{6} + 6\cdot 7^{7} +O\left(7^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 3\cdot 7 + 4\cdot 7^{2} + 7^{3} + 2\cdot 7^{4} + 7^{5} + 7^{6} +O\left(7^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)(2,3)$ |
| $(1,2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,2,4,3)$ |
$\zeta_{4}$ |
$-\zeta_{4}$ |
| $1$ |
$4$ |
$(1,3,4,2)$ |
$-\zeta_{4}$ |
$\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
