Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 a + 48 + \left(44 a + 23\right)\cdot 71 + \left(5 a + 38\right)\cdot 71^{2} + \left(44 a + 52\right)\cdot 71^{3} + \left(62 a + 44\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 59 a + 70 + \left(48 a + 23\right)\cdot 71 + \left(12 a + 4\right)\cdot 71^{2} + \left(a + 40\right)\cdot 71^{3} + \left(46 a + 32\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 59 + 26 a\cdot 71 + \left(65 a + 5\right)\cdot 71^{2} + \left(26 a + 64\right)\cdot 71^{3} + \left(8 a + 54\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 46 + \left(22 a + 62\right)\cdot 71 + \left(58 a + 51\right)\cdot 71^{2} + \left(69 a + 29\right)\cdot 71^{3} + \left(24 a + 52\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 37 + \left(22 a + 7\right)\cdot 71 + \left(18 a + 14\right)\cdot 71^{2} + \left(45 a + 48\right)\cdot 71^{3} + \left(37 a + 34\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 24 + \left(48 a + 23\right)\cdot 71 + \left(52 a + 28\right)\cdot 71^{2} + \left(25 a + 49\right)\cdot 71^{3} + \left(33 a + 64\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)$ |
| $(3,5,4)$ |
| $(1,4,2,5,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,6,2)(3,5,4)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(3,5,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $2$ |
$3$ |
$(3,4,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$-1$ |
$-1$ |
| $3$ |
$6$ |
$(1,4,2,5,6,3)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,3,6,5,2,4)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
