Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 40\cdot 79 + 15\cdot 79^{2} + 69\cdot 79^{3} + 27\cdot 79^{4} + 66\cdot 79^{5} + 40\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 27\cdot 79 + 29\cdot 79^{2} + 8\cdot 79^{3} + 2\cdot 79^{4} + 57\cdot 79^{5} + 64\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 22\cdot 79 + 79^{2} + 61\cdot 79^{3} + 31\cdot 79^{4} + 2\cdot 79^{5} + 40\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 46\cdot 79 + 13\cdot 79^{2} + 46\cdot 79^{3} + 67\cdot 79^{4} + 23\cdot 79^{5} + 61\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 32\cdot 79 + 65\cdot 79^{2} + 32\cdot 79^{3} + 11\cdot 79^{4} + 55\cdot 79^{5} + 17\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 56\cdot 79 + 77\cdot 79^{2} + 17\cdot 79^{3} + 47\cdot 79^{4} + 76\cdot 79^{5} + 38\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 61 + 51\cdot 79 + 49\cdot 79^{2} + 70\cdot 79^{3} + 76\cdot 79^{4} + 21\cdot 79^{5} + 14\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 69 + 38\cdot 79 + 63\cdot 79^{2} + 9\cdot 79^{3} + 51\cdot 79^{4} + 12\cdot 79^{5} + 38\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,5,7,8,3,4,2)$ |
| $(1,8)(4,5)$ |
| $(2,7)(3,6)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $2$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
| $2$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $0$ |
| $2$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
