Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 66\cdot 79 + 33\cdot 79^{2} + 8\cdot 79^{3} + 36\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 19\cdot 79 + 79^{2} + 24\cdot 79^{3} + 20\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 24\cdot 79 + 62\cdot 79^{2} + 73\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 9\cdot 79 + 48\cdot 79^{2} + 37\cdot 79^{3} + 3\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 + 48\cdot 79 + 60\cdot 79^{2} + 51\cdot 79^{3} + 74\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 75\cdot 79 + 65\cdot 79^{2} + 73\cdot 79^{3} + 52\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 + 33\cdot 79 + 15\cdot 79^{2} + 60\cdot 79^{3} + 43\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 + 38\cdot 79 + 28\cdot 79^{2} + 65\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,6)(5,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,4,7)(2,3,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
