Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 9\cdot 79 + 15\cdot 79^{2} + 58\cdot 79^{3} + 66\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 43\cdot 79 + 15\cdot 79^{2} + 69\cdot 79^{3} + 8\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 6\cdot 79 + 67\cdot 79^{2} + 46\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 59\cdot 79 + 36\cdot 79^{2} + 39\cdot 79^{3} + 45\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 19\cdot 79 + 42\cdot 79^{2} + 39\cdot 79^{3} + 33\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 72\cdot 79 + 11\cdot 79^{2} + 32\cdot 79^{3} + 21\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 + 35\cdot 79 + 63\cdot 79^{2} + 9\cdot 79^{3} + 70\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 70 + 69\cdot 79 + 63\cdot 79^{2} + 20\cdot 79^{3} + 12\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7,4,3,8,2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,2,4,6,8,7,5,3)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,5,2,8,3,4,7)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
