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:
| $r_{ 1 }$ |
$=$ |
$ 52 + 57\cdot 67 + 28\cdot 67^{2} + 25\cdot 67^{3} + 15\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 39 + \left(43 a + 25\right)\cdot 67 + \left(18 a + 1\right)\cdot 67^{2} + \left(a + 32\right)\cdot 67^{3} + \left(18 a + 63\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 56 + \left(36 a + 43\right)\cdot 67 + \left(21 a + 37\right)\cdot 67^{2} + \left(43 a + 22\right)\cdot 67^{3} + \left(53 a + 15\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 a + 18 + \left(30 a + 31\right)\cdot 67 + \left(45 a + 20\right)\cdot 67^{2} + \left(23 a + 40\right)\cdot 67^{3} + \left(13 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 a + 51 + \left(23 a + 60\right)\cdot 67 + \left(48 a + 32\right)\cdot 67^{2} + \left(65 a + 18\right)\cdot 67^{3} + 48 a\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 54 + \left(46 a + 8\right)\cdot 67 + \left(62 a + 38\right)\cdot 67^{2} + \left(63 a + 1\right)\cdot 67^{3} + \left(34 a + 56\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 a + \left(20 a + 40\right)\cdot 67 + \left(4 a + 41\right)\cdot 67^{2} + \left(3 a + 60\right)\cdot 67^{3} + \left(32 a + 64\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
