Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 9\cdot 79 + 42\cdot 79^{2} + 18\cdot 79^{3} + 36\cdot 79^{4} + 14\cdot 79^{5} + 55\cdot 79^{6} + 47\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 77\cdot 79 + 44\cdot 79^{2} + 47\cdot 79^{3} + 27\cdot 79^{4} + 55\cdot 79^{5} + 57\cdot 79^{6} + 25\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 4\cdot 79 + 10\cdot 79^{2} + 5\cdot 79^{3} + 73\cdot 79^{4} + 42\cdot 79^{5} + 7\cdot 79^{6} + 13\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 73\cdot 79 + 11\cdot 79^{2} + 6\cdot 79^{3} + 24\cdot 79^{4} + 8\cdot 79^{5} + 17\cdot 79^{6} + 62\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 5\cdot 79 + 67\cdot 79^{2} + 72\cdot 79^{3} + 54\cdot 79^{4} + 70\cdot 79^{5} + 61\cdot 79^{6} + 16\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 74\cdot 79 + 68\cdot 79^{2} + 73\cdot 79^{3} + 5\cdot 79^{4} + 36\cdot 79^{5} + 71\cdot 79^{6} + 65\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 79 + 34\cdot 79^{2} + 31\cdot 79^{3} + 51\cdot 79^{4} + 23\cdot 79^{5} + 21\cdot 79^{6} + 53\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 69\cdot 79 + 36\cdot 79^{2} + 60\cdot 79^{3} + 42\cdot 79^{4} + 64\cdot 79^{5} + 23\cdot 79^{6} + 31\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,5)(3,6)(4,7)$ |
| $(1,7,8,2)(3,4,6,5)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
