Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 78\cdot 79 + 56\cdot 79^{2} + 26\cdot 79^{3} + 67\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 29\cdot 79 + 63\cdot 79^{2} + 64\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 35\cdot 79 + 45\cdot 79^{2} + 19\cdot 79^{3} + 3\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 22\cdot 79 + 31\cdot 79^{2} + 59\cdot 79^{3} + 39\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 20\cdot 79 + 44\cdot 79^{2} + 34\cdot 79^{3} + 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 13\cdot 79 + 58\cdot 79^{2} + 11\cdot 79^{3} + 69\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 21\cdot 79 + 24\cdot 79^{2} + 52\cdot 79^{3} + 47\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 76 + 14\cdot 79 + 71\cdot 79^{2} + 46\cdot 79^{3} + 75\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,6)(3,7)(5,8)$ |
| $(1,2,3,8)(4,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,3,8)(4,5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
