Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 49\cdot 79 + 46\cdot 79^{2} + 76\cdot 79^{3} + 53\cdot 79^{4} + 77\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 5\cdot 79 + 21\cdot 79^{2} + 50\cdot 79^{3} + 27\cdot 79^{4} + 34\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 23\cdot 79 + 25\cdot 79^{2} + 40\cdot 79^{3} + 17\cdot 79^{4} + 42\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 77\cdot 79 + 13\cdot 79^{2} + 9\cdot 79^{3} + 20\cdot 79^{4} + 75\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 79 + 65\cdot 79^{2} + 69\cdot 79^{3} + 58\cdot 79^{4} + 3\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 55\cdot 79 + 53\cdot 79^{2} + 38\cdot 79^{3} + 61\cdot 79^{4} + 36\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 + 73\cdot 79 + 57\cdot 79^{2} + 28\cdot 79^{3} + 51\cdot 79^{4} + 44\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 29\cdot 79 + 32\cdot 79^{2} + 2\cdot 79^{3} + 25\cdot 79^{4} + 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2,5,3)(4,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,3)(4,6,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
