Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 49\cdot 79 + 17\cdot 79^{2} + 30\cdot 79^{3} + 3\cdot 79^{4} + 31\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 71\cdot 79 + 73\cdot 79^{2} + 5\cdot 79^{3} + 12\cdot 79^{4} + 77\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 70\cdot 79 + 49\cdot 79^{2} + 40\cdot 79^{3} + 69\cdot 79^{4} + 13\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 61\cdot 79 + 41\cdot 79^{2} + 19\cdot 79^{3} + 74\cdot 79^{4} + 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 55\cdot 79 + 48\cdot 79^{2} + 67\cdot 79^{3} + 10\cdot 79^{4} + 32\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 + 60\cdot 79 + 17\cdot 79^{2} + 54\cdot 79^{3} + 52\cdot 79^{4} + 17\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 58\cdot 79 + 49\cdot 79^{2} + 16\cdot 79^{3} + 20\cdot 79^{4} + 27\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 45\cdot 79 + 16\cdot 79^{2} + 2\cdot 79^{3} + 73\cdot 79^{4} + 35\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,5)(3,8,7,4)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,8,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
