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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 25\cdot 71 + 16\cdot 71^{2} + 63\cdot 71^{3} + 50\cdot 71^{4} + 56\cdot 71^{5} + 50\cdot 71^{6} + 33\cdot 71^{7} + 52\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 51 + \left(22 a + 9\right)\cdot 71 + \left(25 a + 32\right)\cdot 71^{2} + \left(7 a + 51\right)\cdot 71^{3} + \left(48 a + 4\right)\cdot 71^{4} + \left(9 a + 70\right)\cdot 71^{5} + \left(70 a + 49\right)\cdot 71^{6} + \left(35 a + 67\right)\cdot 71^{7} + \left(66 a + 21\right)\cdot 71^{8} + \left(44 a + 45\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 44 + \left(17 a + 56\right)\cdot 71 + \left(32 a + 65\right)\cdot 71^{2} + \left(5 a + 37\right)\cdot 71^{3} + \left(58 a + 55\right)\cdot 71^{4} + \left(31 a + 8\right)\cdot 71^{5} + \left(70 a + 9\right)\cdot 71^{6} + \left(32 a + 28\right)\cdot 71^{7} + \left(54 a + 12\right)\cdot 71^{8} + \left(26 a + 11\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 14\cdot 71 + 67\cdot 71^{2} + 2\cdot 71^{3} + 55\cdot 71^{4} + 21\cdot 71^{5} + 17\cdot 71^{6} + 61\cdot 71^{7} + 41\cdot 71^{8} + 25\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 a + 50 + \left(53 a + 16\right)\cdot 71 + \left(38 a + 42\right)\cdot 71^{2} + \left(65 a + 16\right)\cdot 71^{3} + \left(12 a + 24\right)\cdot 71^{4} + \left(39 a + 14\right)\cdot 71^{5} + 47\cdot 71^{6} + \left(38 a + 23\right)\cdot 71^{7} + \left(16 a + 17\right)\cdot 71^{8} + \left(44 a + 10\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 50 + \left(48 a + 19\right)\cdot 71 + \left(45 a + 60\right)\cdot 71^{2} + \left(63 a + 40\right)\cdot 71^{3} + \left(22 a + 22\right)\cdot 71^{4} + \left(61 a + 41\right)\cdot 71^{5} + 38\cdot 71^{6} + \left(35 a + 69\right)\cdot 71^{7} + \left(4 a + 47\right)\cdot 71^{8} + \left(26 a + 68\right)\cdot 71^{9} +O\left(71^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2)(3,4,6)$ |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(5,6)$ |
$1$ |
| $3$ |
$2$ |
$(2,3)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,5,2)(3,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(2,5,3,6)$ |
$-1$ |
| $6$ |
$4$ |
$(1,4)(2,5,3,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,3,4,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
