Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 39\cdot 79 + 47\cdot 79^{2} + 14\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 47\cdot 79 + 65\cdot 79^{2} + 42\cdot 79^{3} + 27\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 47\cdot 79 + 75\cdot 79^{2} + 58\cdot 79^{3} + 64\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 30\cdot 79 + 9\cdot 79^{2} + 15\cdot 79^{3} + 42\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 48\cdot 79 + 69\cdot 79^{2} + 63\cdot 79^{3} + 36\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 31\cdot 79 + 3\cdot 79^{2} + 20\cdot 79^{3} + 14\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 31\cdot 79 + 13\cdot 79^{2} + 36\cdot 79^{3} + 51\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 + 39\cdot 79 + 31\cdot 79^{2} + 64\cdot 79^{3} + 21\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,2,5)(4,8,6,7)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,5)(4,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
