Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 7\cdot 59 + 51\cdot 59^{2} + 13\cdot 59^{3} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 27\cdot 59 + 36\cdot 59^{2} + 42\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 38\cdot 59 + 40\cdot 59^{2} + 43\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 55\cdot 59 + 59^{2} + 31\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 30\cdot 59 + 19\cdot 59^{2} + 48\cdot 59^{3} + 3\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 23\cdot 59 + 57\cdot 59^{2} + 42\cdot 59^{3} + 43\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 42\cdot 59 + 39\cdot 59^{2} + 35\cdot 59^{3} + 41\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 42 + 12\cdot 59 + 48\cdot 59^{2} + 36\cdot 59^{3} + 15\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,3,8)(2,4,5,6)$ |
| $(1,4,3,6)(2,8,5,7)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$4$ |
$(1,7,3,8)(2,4,5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,6)(2,8,5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,7,6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
