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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 42 + \left(20 a + 53\right)\cdot 71 + \left(41 a + 47\right)\cdot 71^{2} + \left(26 a + 60\right)\cdot 71^{3} + \left(21 a + 45\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 13\cdot 71 + 55\cdot 71^{2} + 8\cdot 71^{3} + 34\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 31 + \left(49 a + 32\right)\cdot 71 + \left(35 a + 23\right)\cdot 71^{2} + \left(57 a + 50\right)\cdot 71^{3} + \left(2 a + 70\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 12\cdot 71 + 2\cdot 71^{2} + 33\cdot 71^{3} + 52\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 28 + \left(21 a + 26\right)\cdot 71 + \left(35 a + 45\right)\cdot 71^{2} + \left(13 a + 58\right)\cdot 71^{3} + \left(68 a + 18\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 a + 9 + \left(50 a + 4\right)\cdot 71 + \left(29 a + 39\right)\cdot 71^{2} + \left(44 a + 1\right)\cdot 71^{3} + \left(49 a + 62\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,6)$ | $1$ |
| $18$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
| $12$ | $6$ | $(1,2,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
