Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 30\cdot 79 + 49\cdot 79^{2} + 68\cdot 79^{3} + 44\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 62\cdot 79 + 31\cdot 79^{2} + 48\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 35\cdot 79 + 24\cdot 79^{2} + 69\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 9\cdot 79 + 31\cdot 79^{2} + 42\cdot 79^{3} + 61\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 51\cdot 79 + 70\cdot 79^{2} + 76\cdot 79^{3} + 32\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 + 4\cdot 79 + 53\cdot 79^{2} + 56\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 33\cdot 79 + 36\cdot 79^{2} + 53\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 10\cdot 79 + 19\cdot 79^{2} + 32\cdot 79^{3} + 34\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,8)(2,3,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
