Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 30\cdot 79 + 72\cdot 79^{2} + 33\cdot 79^{3} + 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 33\cdot 79 + 31\cdot 79^{2} + 8\cdot 79^{3} + 71\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 61\cdot 79 + 38\cdot 79^{2} + 10\cdot 79^{3} + 71\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 63\cdot 79 + 40\cdot 79^{2} + 69\cdot 79^{3} + 7\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 17\cdot 79 + 67\cdot 79^{2} + 5\cdot 79^{3} + 72\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 + 5\cdot 79 + 43\cdot 79^{2} + 66\cdot 79^{3} + 43\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 76\cdot 79^{2} + 38\cdot 79^{3} + 48\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 24\cdot 79 + 25\cdot 79^{2} + 3\cdot 79^{3} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(3,6)(4,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,2,5,7)(3,8,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,5,2)(3,4,6,8)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,5,3)(2,4,7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,5,7)(3,4,6,8)$ | $0$ |
| $2$ | $4$ | $(1,8,5,4)(2,6,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
