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 + 3\cdot 79^{2} + 48\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 36\cdot 79^{2} + 32\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 26\cdot 79 + 47\cdot 79^{2} + 28\cdot 79^{3} + 22\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 71\cdot 79 + 52\cdot 79^{2} + 44\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 44\cdot 79 + 40\cdot 79^{2} + 50\cdot 79^{3} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 5\cdot 79 + 53\cdot 79^{2} + 28\cdot 79^{3} + 5\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 29\cdot 79 + 22\cdot 79^{2} + 73\cdot 79^{3} + 63\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 78 + 19\cdot 79 + 60\cdot 79^{2} + 9\cdot 79^{3} + 2\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,7,4)(2,6,5,3)$ |
| $(1,7)(2,5)(3,6)(4,8)$ |
| $(1,4,7,8)(2,6,5,3)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,8,7,4)(2,6,5,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,7,8)(2,3,5,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,7,8)(2,6,5,3)$ | $0$ |
| $2$ | $4$ | $(1,6,7,3)(2,8,5,4)$ | $0$ |
| $2$ | $4$ | $(1,2,7,5)(3,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
