Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 26\cdot 79 + 54\cdot 79^{2} + 34\cdot 79^{3} + 70\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 54\cdot 79 + 37\cdot 79^{2} + 73\cdot 79^{3} + 19\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 34\cdot 79 + 64\cdot 79^{2} + 74\cdot 79^{3} + 41\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 5\cdot 79 + 57\cdot 79^{2} + 16\cdot 79^{3} + 61\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 73\cdot 79 + 21\cdot 79^{2} + 62\cdot 79^{3} + 17\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 + 44\cdot 79 + 14\cdot 79^{2} + 4\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 75 + 24\cdot 79 + 41\cdot 79^{2} + 5\cdot 79^{3} + 59\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 78 + 52\cdot 79 + 24\cdot 79^{2} + 44\cdot 79^{3} + 8\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
