Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 2\cdot 79 + 57\cdot 79^{2} + 37\cdot 79^{3} + 23\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 59\cdot 79 + 56\cdot 79^{2} + 14\cdot 79^{3} + 75\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 67\cdot 79 + 68\cdot 79^{2} + 54\cdot 79^{3} + 38\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 49\cdot 79 + 24\cdot 79^{2} + 28\cdot 79^{3} + 58\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 + 29\cdot 79 + 54\cdot 79^{2} + 50\cdot 79^{3} + 20\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 + 11\cdot 79 + 10\cdot 79^{2} + 24\cdot 79^{3} + 40\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 + 19\cdot 79 + 22\cdot 79^{2} + 64\cdot 79^{3} + 3\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 75 + 76\cdot 79 + 21\cdot 79^{2} + 41\cdot 79^{3} + 55\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,8,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
