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:
| $r_{ 1 }$ |
$=$ |
$ 65 a + 49 + \left(29 a + 24\right)\cdot 67 + \left(44 a + 10\right)\cdot 67^{2} + \left(25 a + 60\right)\cdot 67^{3} + \left(31 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 41 + \left(37 a + 12\right)\cdot 67 + \left(22 a + 24\right)\cdot 67^{2} + \left(41 a + 51\right)\cdot 67^{3} + \left(35 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 58\cdot 67 + 31\cdot 67^{2} + 61\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 66\cdot 67 + 45\cdot 67^{2} + 42\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 45 + \left(18 a + 52\right)\cdot 67 + \left(36 a + 47\right)\cdot 67^{2} + \left(16 a + 44\right)\cdot 67^{3} + \left(58 a + 1\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 a + 65 + \left(48 a + 52\right)\cdot 67 + \left(30 a + 40\right)\cdot 67^{2} + \left(50 a + 7\right)\cdot 67^{3} + \left(8 a + 17\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
