Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 34\cdot 71 + 61\cdot 71^{2} + 2\cdot 71^{3} + 70\cdot 71^{4} + 70\cdot 71^{5} + 15\cdot 71^{6} + 7\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 a + 35 + \left(14 a + 58\right)\cdot 71 + \left(17 a + 27\right)\cdot 71^{2} + \left(35 a + 33\right)\cdot 71^{3} + \left(a + 2\right)\cdot 71^{4} + \left(20 a + 57\right)\cdot 71^{5} + \left(7 a + 43\right)\cdot 71^{6} + 50\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 44\cdot 71 + 60\cdot 71^{2} + 38\cdot 71^{3} + 50\cdot 71^{4} + 62\cdot 71^{5} + 67\cdot 71^{6} + 49\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 66 a + 53 + \left(58 a + 28\right)\cdot 71 + \left(57 a + 50\right)\cdot 71^{2} + \left(42 a + 11\right)\cdot 71^{3} + \left(36 a + 9\right)\cdot 71^{4} + \left(16 a + 36\right)\cdot 71^{5} + \left(39 a + 63\right)\cdot 71^{6} + \left(10 a + 39\right)\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 66 + \left(56 a + 36\right)\cdot 71 + \left(53 a + 47\right)\cdot 71^{2} + \left(35 a + 15\right)\cdot 71^{3} + \left(69 a + 41\right)\cdot 71^{4} + \left(50 a + 24\right)\cdot 71^{5} + \left(63 a + 38\right)\cdot 71^{6} + \left(70 a + 43\right)\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 43 + \left(12 a + 9\right)\cdot 71 + \left(13 a + 36\right)\cdot 71^{2} + \left(28 a + 39\right)\cdot 71^{3} + \left(34 a + 39\right)\cdot 71^{4} + \left(54 a + 32\right)\cdot 71^{5} + \left(31 a + 54\right)\cdot 71^{6} + \left(60 a + 21\right)\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,3)$ |
| $(4,6)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(2,5)$ |
$-1$ |
| $4$ |
$3$ |
$(1,4,2)(3,6,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$0$ |
| $4$ |
$6$ |
$(1,6,5,3,4,2)$ |
$0$ |
| $4$ |
$6$ |
$(1,2,4,3,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
