Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 55 a + 27 + \left(13 a + 59\right)\cdot 71 + \left(32 a + 38\right)\cdot 71^{2} + \left(68 a + 67\right)\cdot 71^{3} + \left(53 a + 16\right)\cdot 71^{4} + \left(25 a + 3\right)\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 48 + \left(4 a + 34\right)\cdot 71 + \left(11 a + 54\right)\cdot 71^{2} + \left(4 a + 40\right)\cdot 71^{3} + \left(70 a + 3\right)\cdot 71^{4} + \left(37 a + 26\right)\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 66 + \left(57 a + 31\right)\cdot 71 + \left(38 a + 18\right)\cdot 71^{2} + \left(2 a + 30\right)\cdot 71^{3} + \left(17 a + 56\right)\cdot 71^{4} + 45 a\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 59 a + 35 + \left(15 a + 12\right)\cdot 71 + \left(37 a + 56\right)\cdot 71^{2} + \left(21 a + 50\right)\cdot 71^{3} + \left(10 a + 69\right)\cdot 71^{4} + \left(50 a + 65\right)\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 a + 27 + \left(66 a + 18\right)\cdot 71 + \left(59 a + 1\right)\cdot 71^{2} + \left(66 a + 38\right)\cdot 71^{3} + 68\cdot 71^{4} + \left(33 a + 31\right)\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 11 + \left(55 a + 56\right)\cdot 71 + \left(33 a + 43\right)\cdot 71^{2} + \left(49 a + 56\right)\cdot 71^{3} + \left(60 a + 68\right)\cdot 71^{4} + \left(20 a + 13\right)\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3,4)$ |
| $(1,5,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,3,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,4,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-1$ |
| $3$ | $6$ | $(1,2,6,3,5,4)$ | $0$ |
| $3$ | $6$ | $(1,4,5,3,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
