Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 42\cdot 79 + 69\cdot 79^{2} + 7\cdot 79^{3} + 26\cdot 79^{4} + 71\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 49\cdot 79 + 12\cdot 79^{2} + 14\cdot 79^{3} + 25\cdot 79^{4} + 60\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 48\cdot 79 + 28\cdot 79^{2} + 79^{3} + 77\cdot 79^{4} + 57\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 52\cdot 79 + 2\cdot 79^{2} + 44\cdot 79^{3} + 78\cdot 79^{4} + 3\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 7\cdot 79 + 35\cdot 79^{2} + 19\cdot 79^{3} + 56\cdot 79^{4} + 35\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 12\cdot 79 + 13\cdot 79^{2} + 42\cdot 79^{3} + 28\cdot 79^{4} + 70\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 + 14\cdot 79 + 57\cdot 79^{2} + 25\cdot 79^{3} + 55\cdot 79^{4} + 24\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 78 + 8\cdot 79 + 18\cdot 79^{2} + 3\cdot 79^{3} + 48\cdot 79^{4} + 70\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,3)(2,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,5)(3,4)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,8)(4,6)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,6)(4,7)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,7,8)(2,3,5,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
