Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 5 + 25 a\cdot 71 + \left(60 a + 22\right)\cdot 71^{2} + \left(51 a + 26\right)\cdot 71^{3} + \left(63 a + 38\right)\cdot 71^{4} + \left(70 a + 58\right)\cdot 71^{5} + \left(2 a + 44\right)\cdot 71^{6} + \left(41 a + 65\right)\cdot 71^{7} + \left(32 a + 56\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 48 + \left(41 a + 42\right)\cdot 71 + \left(32 a + 68\right)\cdot 71^{2} + \left(61 a + 17\right)\cdot 71^{3} + \left(12 a + 30\right)\cdot 71^{4} + \left(10 a + 54\right)\cdot 71^{5} + \left(12 a + 33\right)\cdot 71^{6} + \left(23 a + 7\right)\cdot 71^{7} + \left(58 a + 66\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 10 + \left(45 a + 13\right)\cdot 71 + \left(10 a + 46\right)\cdot 71^{2} + \left(19 a + 69\right)\cdot 71^{3} + \left(7 a + 42\right)\cdot 71^{4} + 65\cdot 71^{5} + \left(68 a + 50\right)\cdot 71^{6} + \left(29 a + 2\right)\cdot 71^{7} + \left(38 a + 10\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 66\cdot 71 + 8\cdot 71^{2} + 45\cdot 71^{3} + 51\cdot 71^{4} + 47\cdot 71^{5} + 50\cdot 71^{6} + 56\cdot 71^{7} + 42\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 50\cdot 71 + 45\cdot 71^{2} + 16\cdot 71^{3} + 55\cdot 71^{4} + 66\cdot 71^{5} + 55\cdot 71^{6} + 38\cdot 71^{7} + 19\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 a + 5 + \left(29 a + 40\right)\cdot 71 + \left(38 a + 21\right)\cdot 71^{2} + \left(9 a + 37\right)\cdot 71^{3} + \left(58 a + 65\right)\cdot 71^{4} + \left(60 a + 61\right)\cdot 71^{5} + \left(58 a + 47\right)\cdot 71^{6} + \left(47 a + 41\right)\cdot 71^{7} + \left(12 a + 17\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,6,5)$ |
| $(1,3)$ |
| $(4,5)$ |
| $(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,6)$ | $-1$ |
| $4$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
| $4$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,3,2,4)$ | $0$ |
| $4$ | $6$ | $(1,4,2,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
