Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 12 + \left(36 a + 9\right)\cdot 71 + \left(18 a + 48\right)\cdot 71^{2} + \left(36 a + 21\right)\cdot 71^{3} + \left(56 a + 40\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 27 + \left(31 a + 33\right)\cdot 71 + \left(33 a + 11\right)\cdot 71^{2} + \left(18 a + 7\right)\cdot 71^{3} + \left(68 a + 18\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 19\cdot 71 + 70\cdot 71^{2} + 45\cdot 71^{3} + 14\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 20 + \left(39 a + 64\right)\cdot 71 + \left(37 a + 46\right)\cdot 71^{2} + \left(52 a + 10\right)\cdot 71^{3} + \left(2 a + 65\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 17\cdot 71 + 58\cdot 71^{2} + 51\cdot 71^{3} + 28\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 a + 36 + \left(34 a + 69\right)\cdot 71 + \left(52 a + 48\right)\cdot 71^{2} + \left(34 a + 4\right)\cdot 71^{3} + \left(14 a + 46\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,5)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,5,6)$ | $1$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
| $12$ | $6$ | $(1,5,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
