Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 68\cdot 79 + 28\cdot 79^{2} + 24\cdot 79^{3} + 9\cdot 79^{4} + 39\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 47\cdot 79 + 74\cdot 79^{2} + 30\cdot 79^{3} + 27\cdot 79^{4} + 48\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 22\cdot 79 + 51\cdot 79^{2} + 61\cdot 79^{3} + 34\cdot 79^{4} + 34\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 58\cdot 79 + 75\cdot 79^{2} + 37\cdot 79^{3} + 71\cdot 79^{4} + 42\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 20\cdot 79 + 3\cdot 79^{2} + 41\cdot 79^{3} + 7\cdot 79^{4} + 36\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 56\cdot 79 + 27\cdot 79^{2} + 17\cdot 79^{3} + 44\cdot 79^{4} + 44\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 31\cdot 79 + 4\cdot 79^{2} + 48\cdot 79^{3} + 51\cdot 79^{4} + 30\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 76 + 10\cdot 79 + 50\cdot 79^{2} + 54\cdot 79^{3} + 69\cdot 79^{4} + 39\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2,5,3)(4,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,5,3)(4,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
