Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{3} + 3 x + 51 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 a^{2} + 44 a + 6 + \left(10 a^{2} + 11 a + 20\right)\cdot 53 + \left(13 a^{2} + 22 a + 2\right)\cdot 53^{2} + \left(23 a^{2} + 45 a + 51\right)\cdot 53^{3} + \left(25 a^{2} + 31 a + 13\right)\cdot 53^{4} + \left(21 a^{2} + 25 a + 21\right)\cdot 53^{5} + \left(43 a^{2} + 50 a + 8\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 45\cdot 53 + 4\cdot 53^{2} + 46\cdot 53^{3} + 48\cdot 53^{4} + 24\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 34\cdot 53 + 11\cdot 53^{2} + 37\cdot 53^{3} + 53^{4} + 2\cdot 53^{5} + 43\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 38\cdot 53 + 32\cdot 53^{2} + 24\cdot 53^{3} + 28\cdot 53^{4} + 47\cdot 53^{5} + 35\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a^{2} + 47 a + 18 + \left(9 a^{2} + 36 a + 18\right)\cdot 53 + \left(12 a^{2} + 31 a\right)\cdot 53^{2} + \left(26 a + 5\right)\cdot 53^{3} + \left(9 a^{2} + 46 a + 34\right)\cdot 53^{4} + \left(17 a^{2} + 11 a + 12\right)\cdot 53^{5} + \left(23 a^{2} + 21 a + 21\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a^{2} + 51 a + 3 + \left(3 a^{2} + 2 a + 4\right)\cdot 53 + \left(13 a^{2} + 10 a + 16\right)\cdot 53^{2} + \left(14 a^{2} + 18 a + 23\right)\cdot 53^{3} + \left(4 a^{2} + 9 a + 1\right)\cdot 53^{4} + \left(28 a^{2} + 52 a + 8\right)\cdot 53^{5} + \left(20 a^{2} + 37 a + 32\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 14 a^{2} + 42 a + 26 + \left(38 a^{2} + 38 a + 20\right)\cdot 53 + \left(11 a^{2} + 4 a + 13\right)\cdot 53^{2} + \left(50 a^{2} + 40 a + 42\right)\cdot 53^{3} + \left(2 a^{2} + 43 a + 51\right)\cdot 53^{4} + \left(14 a^{2} + 19 a + 32\right)\cdot 53^{5} + \left(23 a^{2} + 46 a + 37\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 48 a^{2} + 15 a + 50 + \left(32 a^{2} + 4 a + 11\right)\cdot 53 + \left(27 a^{2} + 52 a + 31\right)\cdot 53^{2} + \left(29 a^{2} + 33 a + 10\right)\cdot 53^{3} + \left(18 a^{2} + 27 a\right)\cdot 53^{4} + \left(14 a^{2} + 15 a + 7\right)\cdot 53^{5} + \left(39 a^{2} + 34 a\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 10 a^{2} + 13 a + 18 + \left(11 a^{2} + 11 a + 19\right)\cdot 53 + \left(28 a^{2} + 38 a + 46\right)\cdot 53^{2} + \left(41 a^{2} + 47 a + 24\right)\cdot 53^{3} + \left(45 a^{2} + 52 a + 31\right)\cdot 53^{4} + \left(10 a^{2} + 33 a + 26\right)\cdot 53^{5} + \left(9 a^{2} + 21 a + 9\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,6,8,4,7,5,3,9)$ |
| $(2,4,3)(6,9,7)$ |
| $(1,8,5)(2,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $3$ | $(1,8,5)(2,4,3)(6,7,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,8)(2,3,4)(6,9,7)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,8,5)(2,3,4)$ | $0$ |
| $3$ | $3$ | $(1,5,8)(2,4,3)$ | $0$ |
| $3$ | $9$ | $(1,2,6,8,4,7,5,3,9)$ | $0$ |
| $3$ | $9$ | $(1,6,4,5,9,2,8,7,3)$ | $0$ |
| $3$ | $9$ | $(1,3,9,8,2,6,5,4,7)$ | $0$ |
| $3$ | $9$ | $(1,9,2,5,7,3,8,6,4)$ | $0$ |
| $3$ | $9$ | $(1,7,3,5,6,4,8,9,2)$ | $0$ |
| $3$ | $9$ | $(1,3,6,8,2,7,5,4,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
