Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 43\cdot 79 + 24\cdot 79^{2} + 4\cdot 79^{3} + 13\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 4\cdot 79 + 56\cdot 79^{2} + 36\cdot 79^{3} + 63\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 15\cdot 79 + 18\cdot 79^{2} + 67\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 24\cdot 79 + 33\cdot 79^{2} + 45\cdot 79^{3} + 78\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 37\cdot 79 + 72\cdot 79^{2} + 50\cdot 79^{3} + 56\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 51\cdot 79 + 20\cdot 79^{2} + 25\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 63 + 30\cdot 79 + 79^{2} + 12\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 + 29\cdot 79 + 10\cdot 79^{2} + 62\cdot 79^{3} + 10\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,6)(3,5)(7,8)$ |
| $(1,2,4,6)(3,8,5,7)$ |
| $(1,8,4,7)(2,3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)(7,8)$ |
$-2$ |
| $2$ |
$4$ |
$(1,8,4,7)(2,3,6,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,6)(3,8,5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,4,3)(2,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
