Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{3} + 3 x + 51 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 a^{2} + 41 a + 13 + \left(48 a^{2} + 47 a + 44\right)\cdot 53 + \left(37 a^{2} + 47 a + 28\right)\cdot 53^{2} + \left(20 a^{2} + 51 a + 42\right)\cdot 53^{3} + \left(34 a^{2} + 27 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a^{2} + 41 a + 31 + \left(11 a^{2} + 47 a + 39\right)\cdot 53 + \left(40 a + 13\right)\cdot 53^{2} + \left(5 a^{2} + 29 a + 29\right)\cdot 53^{3} + \left(38 a^{2} + 2 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a^{2} + 7 a + 34 + \left(48 a^{2} + 42 a + 43\right)\cdot 53 + \left(44 a^{2} + 23 a + 42\right)\cdot 53^{2} + \left(37 a^{2} + 30 a + 23\right)\cdot 53^{3} + \left(25 a^{2} + 39 a + 21\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 a^{2} + 50 a + 33 + \left(36 a^{2} + 10 a + 38\right)\cdot 53 + \left(31 a^{2} + 38 a + 8\right)\cdot 53^{2} + \left(12 a^{2} + 25 a + 40\right)\cdot 53^{3} + \left(a^{2} + 2 a + 12\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a^{2} + 17 a + 8 + \left(24 a^{2} + 42 a + 13\right)\cdot 53 + \left(30 a^{2} + 25 a + 6\right)\cdot 53^{2} + \left(10 a^{2} + 38 a + 36\right)\cdot 53^{3} + \left(6 a^{2} + 25 a + 22\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a^{2} + 39 a + 24 + \left(45 a^{2} + 52 a + 2\right)\cdot 53 + \left(43 a^{2} + 41 a + 33\right)\cdot 53^{2} + \left(29 a^{2} + 41 a + 21\right)\cdot 53^{3} + \left(45 a^{2} + 24 a + 48\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 a^{2} + 21 a + 39 + \left(10 a^{2} + 52 a + 37\right)\cdot 53 + \left(29 a^{2} + 31 a + 18\right)\cdot 53^{2} + \left(3 a^{2} + 13 a + 26\right)\cdot 53^{3} + \left(2 a^{2} + 32 a + 41\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 23 a^{2} + 5 a + 13 + \left(9 a^{2} + 16 a + 19\right)\cdot 53 + \left(23 a^{2} + 34 a + 52\right)\cdot 53^{2} + \left(47 a^{2} + 23 a + 42\right)\cdot 53^{3} + \left(45 a^{2} + 38 a + 8\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 13 a^{2} + 44 a + 21 + \left(31 a^{2} + 5 a + 26\right)\cdot 53 + \left(23 a^{2} + 33 a + 7\right)\cdot 53^{2} + \left(44 a^{2} + 9 a + 2\right)\cdot 53^{3} + \left(12 a^{2} + 18 a + 10\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5)(2,9)(3,4)(6,8)$ |
| $(1,2,6,8,9,5,3,7,4)$ |
| $(1,8,3)(2,9,7)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,5)(2,9)(3,4)(6,8)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,8,3)(2,9,7)(4,6,5)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,2,6,8,9,5,3,7,4)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,6,9,3,4,2,8,5,7)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,9,4,8,7,6,3,2,5)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
