Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 36\cdot 67 + 39\cdot 67^{2} + 52\cdot 67^{3} + 49\cdot 67^{4} + 10\cdot 67^{5} + 40\cdot 67^{6} + 14\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 + 30\cdot 67 + 27\cdot 67^{2} + 14\cdot 67^{3} + 17\cdot 67^{4} + 56\cdot 67^{5} + 26\cdot 67^{6} + 52\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 18 + \left(61 a + 49\right)\cdot 67 + \left(11 a + 6\right)\cdot 67^{2} + \left(63 a + 47\right)\cdot 67^{3} + \left(3 a + 23\right)\cdot 67^{4} + \left(55 a + 59\right)\cdot 67^{5} + \left(2 a + 21\right)\cdot 67^{6} + \left(36 a + 63\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 35 + \left(50 a + 49\right)\cdot 67 + \left(8 a + 7\right)\cdot 67^{2} + \left(16 a + 39\right)\cdot 67^{3} + \left(30 a + 14\right)\cdot 67^{4} + \left(61 a + 26\right)\cdot 67^{5} + \left(54 a + 21\right)\cdot 67^{6} + \left(49 a + 28\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 50 + \left(5 a + 17\right)\cdot 67 + \left(55 a + 60\right)\cdot 67^{2} + \left(3 a + 19\right)\cdot 67^{3} + \left(63 a + 43\right)\cdot 67^{4} + \left(11 a + 7\right)\cdot 67^{5} + \left(64 a + 45\right)\cdot 67^{6} + \left(30 a + 3\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 33 + \left(16 a + 17\right)\cdot 67 + \left(58 a + 59\right)\cdot 67^{2} + \left(50 a + 27\right)\cdot 67^{3} + \left(36 a + 52\right)\cdot 67^{4} + \left(5 a + 40\right)\cdot 67^{5} + \left(12 a + 45\right)\cdot 67^{6} + \left(17 a + 38\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,4,3)(2,6,5)$ |
| $(1,4)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $8$ | $3$ | $(1,4,3)(2,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,2,5)$ | $1$ |
| $6$ | $4$ | $(1,2)(3,6,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,4,3,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
