Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 32\cdot 79 + 64\cdot 79^{2} + 77\cdot 79^{3} + 43\cdot 79^{4} + 47\cdot 79^{6} + 15\cdot 79^{7} + 32\cdot 79^{8} + 40\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 46\cdot 79 + 52\cdot 79^{2} + 44\cdot 79^{3} + 53\cdot 79^{4} + 15\cdot 79^{5} + 20\cdot 79^{6} + 65\cdot 79^{7} + 62\cdot 79^{8} + 51\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 54\cdot 79 + 64\cdot 79^{2} + 18\cdot 79^{3} + 27\cdot 79^{4} + 18\cdot 79^{5} + 43\cdot 79^{6} + 29\cdot 79^{7} + 61\cdot 79^{8} + 64\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 61\cdot 79 + 76\cdot 79^{2} + 61\cdot 79^{3} + 45\cdot 79^{4} + 75\cdot 79^{5} + 37\cdot 79^{6} + 66\cdot 79^{7} + 45\cdot 79^{8} + 58\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 + 17\cdot 79 + 2\cdot 79^{2} + 17\cdot 79^{3} + 33\cdot 79^{4} + 3\cdot 79^{5} + 41\cdot 79^{6} + 12\cdot 79^{7} + 33\cdot 79^{8} + 20\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 24\cdot 79 + 14\cdot 79^{2} + 60\cdot 79^{3} + 51\cdot 79^{4} + 60\cdot 79^{5} + 35\cdot 79^{6} + 49\cdot 79^{7} + 17\cdot 79^{8} + 14\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 32\cdot 79 + 26\cdot 79^{2} + 34\cdot 79^{3} + 25\cdot 79^{4} + 63\cdot 79^{5} + 58\cdot 79^{6} + 13\cdot 79^{7} + 16\cdot 79^{8} + 27\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 46\cdot 79 + 14\cdot 79^{2} + 79^{3} + 35\cdot 79^{4} + 78\cdot 79^{5} + 31\cdot 79^{6} + 63\cdot 79^{7} + 46\cdot 79^{8} + 38\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
