Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 37 + \left(26 a + 28\right)\cdot 67 + \left(16 a + 65\right)\cdot 67^{2} + \left(44 a + 12\right)\cdot 67^{3} + \left(50 a + 53\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 46\cdot 67 + 9\cdot 67^{2} + 50\cdot 67^{3} + 19\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 32\cdot 67 + 33\cdot 67^{2} + 43\cdot 67^{3} + 43\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 a + 45 + \left(5 a + 24\right)\cdot 67 + \left(54 a + 56\right)\cdot 67^{2} + \left(49 a + 21\right)\cdot 67^{3} + \left(12 a + 36\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 36 + \left(40 a + 18\right)\cdot 67 + \left(50 a + 37\right)\cdot 67^{2} + \left(22 a + 39\right)\cdot 67^{3} + \left(16 a + 10\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 37 + \left(61 a + 50\right)\cdot 67 + \left(12 a + 65\right)\cdot 67^{2} + \left(17 a + 32\right)\cdot 67^{3} + \left(54 a + 37\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3,2,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $15$ | $2$ | $(1,5)(4,6)$ | $1$ |
| $20$ | $3$ | $(1,2,6)(3,4,5)$ | $-1$ |
| $30$ | $4$ | $(1,6,5,4)$ | $-1$ |
| $24$ | $5$ | $(1,4,2,3,6)$ | $0$ |
| $20$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
