Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a^{2} + 15 a + 34 + \left(49 a^{2} + 12 a + 26\right)\cdot 61 + \left(39 a^{2} + 19 a + 43\right)\cdot 61^{2} + \left(10 a^{2} + 39 a + 49\right)\cdot 61^{3} + \left(58 a^{2} + 57 a + 47\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a^{2} + 37 a + 31 + \left(15 a^{2} + 42 a + 50\right)\cdot 61 + \left(44 a^{2} + 39 a + 43\right)\cdot 61^{2} + \left(55 a^{2} + 45 a + 56\right)\cdot 61^{3} + \left(39 a^{2} + 31 a + 43\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a^{2} + 13 a + 1 + \left(37 a^{2} + 42 a + 32\right)\cdot 61 + \left(54 a^{2} + 39 a + 51\right)\cdot 61^{2} + \left(49 a^{2} + 44 a + 49\right)\cdot 61^{3} + \left(55 a^{2} + 43 a + 36\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a^{2} + 34 a + 12 + \left(37 a^{2} + 33 a + 52\right)\cdot 61 + \left(49 a^{2} + 15 a + 7\right)\cdot 61^{2} + \left(42 a^{2} + 50 a + 37\right)\cdot 61^{3} + \left(9 a^{2} + 17 a + 4\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a^{2} + 31 a + 54 + \left(39 a^{2} + 51 a + 41\right)\cdot 61 + \left(31 a^{2} + 54 a + 25\right)\cdot 61^{2} + \left(47 a^{2} + 36 a + 18\right)\cdot 61^{3} + \left(49 a^{2} + 49\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a^{2} + a + 47 + \left(41 a^{2} + 34 a + 12\right)\cdot 61 + \left(43 a^{2} + 57 a + 41\right)\cdot 61^{2} + \left(23 a^{2} + 35 a + 49\right)\cdot 61^{3} + \left(24 a^{2} + 15 a + 52\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 a^{2} + 12 a + 15 + \left(35 a^{2} + 15 a + 43\right)\cdot 61 + \left(32 a^{2} + 26 a + 9\right)\cdot 61^{2} + \left(7 a^{2} + 32 a + 35\right)\cdot 61^{3} + \left(54 a^{2} + 46 a + 8\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 21 a^{2} + 54 a + 37 + \left(6 a^{2} + 27 a + 29\right)\cdot 61 + \left(46 a^{2} + 27 a + 52\right)\cdot 61^{2} + \left(18 a^{2} + 39 a + 46\right)\cdot 61^{3} + \left(32 a^{2} + 28 a + 28\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 42 a^{2} + 47 a + 13 + \left(42 a^{2} + 45 a + 16\right)\cdot 61 + \left(23 a^{2} + 24 a + 29\right)\cdot 61^{2} + \left(48 a^{2} + 41 a + 22\right)\cdot 61^{3} + \left(41 a^{2} + a + 32\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,9,7,5,6,4,2,3)$ |
| $(1,6)(3,4)(5,8)(7,9)$ |
| $(1,7,4)(2,8,5)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,6)(3,4)(5,8)(7,9)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,7,4)(2,8,5)(3,9,6)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,8,9,7,5,6,4,2,3)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,9,5,4,3,8,7,6,2)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,5,3,7,2,9,4,8,6)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
