Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 57 + \left(27 a + 34\right)\cdot 59 + \left(22 a + 21\right)\cdot 59^{2} + \left(17 a + 51\right)\cdot 59^{3} + \left(7 a + 31\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 52\cdot 59 + 24\cdot 59^{2} + 43\cdot 59^{3} + 21\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 a + 7 + \left(31 a + 53\right)\cdot 59 + \left(36 a + 16\right)\cdot 59^{2} + \left(41 a + 46\right)\cdot 59^{3} + \left(51 a + 21\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 50 + \left(28 a + 9\right)\cdot 59 + \left(39 a + 10\right)\cdot 59^{2} + \left(11 a + 51\right)\cdot 59^{3} + \left(28 a + 32\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 a + 54 + \left(30 a + 33\right)\cdot 59 + \left(19 a + 21\right)\cdot 59^{2} + \left(47 a + 23\right)\cdot 59^{3} + \left(30 a + 49\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 51\cdot 59 + 22\cdot 59^{2} + 20\cdot 59^{3} + 19\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4,3,5,2)$ |
| $(1,4)(2,5)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $15$ | $2$ | $(1,6)(2,5)$ | $1$ |
| $20$ | $3$ | $(1,4,5)(2,6,3)$ | $-1$ |
| $30$ | $4$ | $(1,5,6,2)$ | $1$ |
| $24$ | $5$ | $(1,6,4,2,3)$ | $0$ |
| $20$ | $6$ | $(1,6,4,3,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
