Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 26\cdot 71 + 35\cdot 71^{2} + 20\cdot 71^{3} + 26\cdot 71^{4} + 4\cdot 71^{5} + 6\cdot 71^{6} + 29\cdot 71^{7} + 48\cdot 71^{8} + 19\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 42 + \left(56 a + 14\right)\cdot 71 + \left(57 a + 15\right)\cdot 71^{2} + \left(61 a + 49\right)\cdot 71^{3} + \left(33 a + 50\right)\cdot 71^{4} + \left(63 a + 9\right)\cdot 71^{5} + \left(39 a + 46\right)\cdot 71^{6} + \left(52 a + 13\right)\cdot 71^{7} + \left(57 a + 22\right)\cdot 71^{8} + \left(16 a + 23\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 a + 28 + \left(56 a + 50\right)\cdot 71 + \left(57 a + 67\right)\cdot 71^{2} + \left(61 a + 26\right)\cdot 71^{3} + \left(33 a + 14\right)\cdot 71^{4} + \left(63 a + 39\right)\cdot 71^{5} + \left(39 a + 8\right)\cdot 71^{6} + \left(52 a + 63\right)\cdot 71^{7} + \left(57 a + 56\right)\cdot 71^{8} + 16 a\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 + 44\cdot 71 + 35\cdot 71^{2} + 50\cdot 71^{3} + 44\cdot 71^{4} + 66\cdot 71^{5} + 64\cdot 71^{6} + 41\cdot 71^{7} + 22\cdot 71^{8} + 51\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 29 + \left(14 a + 56\right)\cdot 71 + \left(13 a + 55\right)\cdot 71^{2} + \left(9 a + 21\right)\cdot 71^{3} + \left(37 a + 20\right)\cdot 71^{4} + \left(7 a + 61\right)\cdot 71^{5} + \left(31 a + 24\right)\cdot 71^{6} + \left(18 a + 57\right)\cdot 71^{7} + \left(13 a + 48\right)\cdot 71^{8} + \left(54 a + 47\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 43 + \left(14 a + 20\right)\cdot 71 + \left(13 a + 3\right)\cdot 71^{2} + \left(9 a + 44\right)\cdot 71^{3} + \left(37 a + 56\right)\cdot 71^{4} + \left(7 a + 31\right)\cdot 71^{5} + \left(31 a + 62\right)\cdot 71^{6} + \left(18 a + 7\right)\cdot 71^{7} + \left(13 a + 14\right)\cdot 71^{8} + \left(54 a + 70\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(4,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,4,6)$ |
$1$ |
| $6$ |
$4$ |
$(1,4)(2,3,5,6)$ |
$-1$ |
| $8$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
