Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 36\cdot 79 + 66\cdot 79^{2} + 18\cdot 79^{3} + 4\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 7\cdot 79 + 36\cdot 79^{2} + 65\cdot 79^{3} + 51\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 5\cdot 79 + 44\cdot 79^{2} + 68\cdot 79^{3} + 18\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 54\cdot 79 + 9\cdot 79^{2} + 5\cdot 79^{3} + 5\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 24\cdot 79 + 69\cdot 79^{2} + 73\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 73\cdot 79 + 34\cdot 79^{2} + 10\cdot 79^{3} + 60\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 62 + 71\cdot 79 + 42\cdot 79^{2} + 13\cdot 79^{3} + 27\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 42\cdot 79 + 12\cdot 79^{2} + 60\cdot 79^{3} + 74\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
