Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 65 a + 58 + \left(2 a + 19\right)\cdot 71 + \left(36 a + 26\right)\cdot 71^{2} + \left(70 a + 6\right)\cdot 71^{3} + \left(10 a + 2\right)\cdot 71^{4} + \left(57 a + 55\right)\cdot 71^{5} + \left(52 a + 3\right)\cdot 71^{6} + \left(66 a + 18\right)\cdot 71^{7} + \left(3 a + 24\right)\cdot 71^{8} + \left(19 a + 44\right)\cdot 71^{9} + \left(2 a + 45\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 65 a + 25 + \left(2 a + 39\right)\cdot 71 + \left(36 a + 46\right)\cdot 71^{2} + \left(70 a + 30\right)\cdot 71^{3} + \left(10 a + 46\right)\cdot 71^{4} + \left(57 a + 54\right)\cdot 71^{5} + \left(52 a + 18\right)\cdot 71^{6} + \left(66 a + 43\right)\cdot 71^{7} + \left(3 a + 34\right)\cdot 71^{8} + \left(19 a + 63\right)\cdot 71^{9} + \left(2 a + 39\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 28\cdot 71 + 43\cdot 71^{2} + 24\cdot 71^{3} + 39\cdot 71^{4} + 63\cdot 71^{5} + 51\cdot 71^{6} + 4\cdot 71^{7} + 10\cdot 71^{8} + 61\cdot 71^{9} + 18\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 13 + \left(68 a + 51\right)\cdot 71 + \left(34 a + 44\right)\cdot 71^{2} + 64\cdot 71^{3} + \left(60 a + 68\right)\cdot 71^{4} + \left(13 a + 15\right)\cdot 71^{5} + \left(18 a + 67\right)\cdot 71^{6} + \left(4 a + 52\right)\cdot 71^{7} + \left(67 a + 46\right)\cdot 71^{8} + \left(51 a + 26\right)\cdot 71^{9} + \left(68 a + 25\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 46 + \left(68 a + 31\right)\cdot 71 + \left(34 a + 24\right)\cdot 71^{2} + 40\cdot 71^{3} + \left(60 a + 24\right)\cdot 71^{4} + \left(13 a + 16\right)\cdot 71^{5} + \left(18 a + 52\right)\cdot 71^{6} + \left(4 a + 27\right)\cdot 71^{7} + \left(67 a + 36\right)\cdot 71^{8} + \left(51 a + 7\right)\cdot 71^{9} + \left(68 a + 31\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 42\cdot 71 + 27\cdot 71^{2} + 46\cdot 71^{3} + 31\cdot 71^{4} + 7\cdot 71^{5} + 19\cdot 71^{6} + 66\cdot 71^{7} + 60\cdot 71^{8} + 9\cdot 71^{9} + 52\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,4,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,4)(2,6,5,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,6,5,4,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
