Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 51\cdot 79 + 30\cdot 79^{2} + 59\cdot 79^{3} + 25\cdot 79^{4} + 64\cdot 79^{5} + 72\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 66\cdot 79 + 72\cdot 79^{2} + 19\cdot 79^{4} + 60\cdot 79^{5} + 7\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 57\cdot 79 + 61\cdot 79^{2} + 31\cdot 79^{3} + 68\cdot 79^{4} + 16\cdot 79^{5} + 6\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 79 + 73\cdot 79^{2} + 57\cdot 79^{3} + 6\cdot 79^{4} + 43\cdot 79^{5} + 75\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 72\cdot 79 + 24\cdot 79^{2} + 52\cdot 79^{3} + 61\cdot 79^{4} + 12\cdot 79^{5} + 20\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 33\cdot 79 + 29\cdot 79^{2} + 67\cdot 79^{3} + 63\cdot 79^{4} + 37\cdot 79^{5} + 68\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 5\cdot 79 + 58\cdot 79^{2} + 26\cdot 79^{3} + 26\cdot 79^{4} + 3\cdot 79^{5} + 56\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 74 + 28\cdot 79 + 44\cdot 79^{2} + 19\cdot 79^{3} + 44\cdot 79^{4} + 77\cdot 79^{5} + 8\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,6)(2,8,3,7)$ |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,5,6)(2,8,3,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
