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:
| $r_{ 1 }$ |
$=$ |
$ 61 a + 57 + \left(64 a + 11\right)\cdot 71 + \left(38 a + 65\right)\cdot 71^{2} + \left(22 a + 36\right)\cdot 71^{3} + \left(30 a + 56\right)\cdot 71^{4} + \left(52 a + 11\right)\cdot 71^{5} + \left(13 a + 33\right)\cdot 71^{6} + \left(42 a + 62\right)\cdot 71^{7} + \left(57 a + 34\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 12 + \left(69 a + 46\right)\cdot 71 + \left(13 a + 56\right)\cdot 71^{2} + \left(3 a + 56\right)\cdot 71^{3} + \left(a + 24\right)\cdot 71^{4} + \left(26 a + 49\right)\cdot 71^{5} + \left(35 a + 64\right)\cdot 71^{6} + \left(12 a + 35\right)\cdot 71^{7} + \left(58 a + 47\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 + 35\cdot 71 + 47\cdot 71^{2} + 41\cdot 71^{3} + 41\cdot 71^{4} + 66\cdot 71^{5} + 24\cdot 71^{6} + 11\cdot 71^{7} + 61\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 37 + \left(6 a + 9\right)\cdot 71 + \left(32 a + 7\right)\cdot 71^{2} + \left(48 a + 43\right)\cdot 71^{3} + \left(40 a + 23\right)\cdot 71^{4} + \left(18 a + 15\right)\cdot 71^{5} + \left(57 a + 8\right)\cdot 71^{6} + \left(28 a + 62\right)\cdot 71^{7} + \left(13 a + 36\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 3 + \left(a + 12\right)\cdot 71 + \left(57 a + 15\right)\cdot 71^{2} + \left(67 a + 49\right)\cdot 71^{3} + \left(69 a + 23\right)\cdot 71^{4} + \left(44 a + 29\right)\cdot 71^{5} + \left(35 a + 38\right)\cdot 71^{6} + \left(58 a + 25\right)\cdot 71^{7} + \left(12 a + 9\right)\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 26\cdot 71 + 21\cdot 71^{2} + 56\cdot 71^{3} + 42\cdot 71^{4} + 40\cdot 71^{5} + 43\cdot 71^{6} + 15\cdot 71^{7} + 23\cdot 71^{8} +O\left(71^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,5)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $3$ | $2$ | $(1,5)(2,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,5,2)$ | $-1$ |
| $6$ | $4$ | $(1,5)(2,6,4,3)$ | $1$ |
| $8$ | $6$ | $(1,4,6,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
