Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 67\cdot 79 + 24\cdot 79^{2} + 24\cdot 79^{3} + 72\cdot 79^{4} + 4\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 4\cdot 79^{2} + 61\cdot 79^{3} + 64\cdot 79^{4} + 5\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 74\cdot 79 + 16\cdot 79^{2} + 11\cdot 79^{3} + 39\cdot 79^{4} + 25\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 56\cdot 79 + 30\cdot 79^{2} + 8\cdot 79^{3} + 15\cdot 79^{4} + 72\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 39\cdot 79 + 6\cdot 79^{2} + 35\cdot 79^{3} + 31\cdot 79^{4} + 55\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 34\cdot 79 + 19\cdot 79^{2} + 64\cdot 79^{3} + 5\cdot 79^{4} + 75\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 20\cdot 79 + 7\cdot 79^{2} + 31\cdot 79^{3} + 12\cdot 79^{4} + 75\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 62 + 23\cdot 79 + 48\cdot 79^{2} + 79^{3} + 75\cdot 79^{4} + 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)(2,7,6,8)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,5)(2,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
