Basic invariants
Dimension: | $3$ |
Group: | 12T29 |
Conductor: | \(561125\)\(\medspace = 5^{3} \cdot 67^{2} \) |
Artin stem field: | Galois closure of 12.8.793100932727814453125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 12T29 |
Parity: | odd |
Determinant: | 1.5.4t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.112225.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} - 32 x^{10} + 5 x^{9} + 171 x^{8} - 55 x^{7} + 1092 x^{6} + 368 x^{5} - 5094 x^{4} + \cdots - 2305 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{4} + 5x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 41 a^{3} + 39 a^{2} + 3 a + 40 + \left(a^{3} + 12 a^{2} + 29 a + 39\right)\cdot 43 + \left(19 a^{3} + 15 a^{2} + 28 a + 30\right)\cdot 43^{2} + \left(39 a^{3} + 21 a^{2} + 15 a + 30\right)\cdot 43^{3} + \left(22 a^{3} + 35 a^{2} + 33 a + 28\right)\cdot 43^{4} + \left(6 a^{3} + 31 a^{2} + 13 a + 32\right)\cdot 43^{5} + \left(32 a^{3} + 35 a^{2} + 10 a + 19\right)\cdot 43^{6} + \left(36 a^{3} + 31 a^{2} + 20 a + 1\right)\cdot 43^{7} + \left(15 a^{3} + 3 a^{2} + 29 a + 30\right)\cdot 43^{8} + \left(27 a^{3} + 29 a^{2} + 35 a + 15\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 2 }$ | $=$ | \( 11 a^{3} + 16 a^{2} + 7 a + 25 + \left(24 a^{3} + 22 a^{2} + 23 a + 41\right)\cdot 43 + \left(9 a^{3} + 32 a^{2} + 16 a + 25\right)\cdot 43^{2} + \left(13 a^{3} + 41 a^{2} + 35 a + 5\right)\cdot 43^{3} + \left(7 a^{3} + 22 a^{2} + 23 a + 18\right)\cdot 43^{4} + \left(29 a^{3} + 14 a^{2} + 22 a + 31\right)\cdot 43^{5} + \left(4 a^{3} + 34 a^{2} + 6 a + 2\right)\cdot 43^{6} + \left(18 a^{3} + 9 a^{2} + 31 a + 35\right)\cdot 43^{7} + \left(41 a^{3} + 28 a^{2} + 3 a + 24\right)\cdot 43^{8} + \left(25 a^{3} + 5 a^{2} + 32 a + 31\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 3 }$ | $=$ | \( 5 a^{3} + 23 a^{2} + 36 a + 23 + \left(37 a^{3} + 20 a^{2} + 18 a + 42\right)\cdot 43 + \left(39 a^{3} + 20 a^{2} + 18 a + 23\right)\cdot 43^{2} + \left(23 a^{3} + 37 a + 30\right)\cdot 43^{3} + \left(23 a^{3} + 21 a^{2} + 3 a + 7\right)\cdot 43^{4} + \left(19 a^{3} + 24 a^{2} + 8 a + 15\right)\cdot 43^{5} + \left(11 a^{3} + 28 a^{2} + 5 a + 14\right)\cdot 43^{6} + \left(11 a^{3} + 13 a^{2} + 2\right)\cdot 43^{7} + \left(30 a^{3} + 28 a^{2} + 7 a + 26\right)\cdot 43^{8} + \left(9 a^{3} + 13 a^{2} + 24 a + 27\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 4 }$ | $=$ | \( 22 a^{3} + 33 a^{2} + 39 a + 7 + \left(9 a^{3} + 13 a^{2} + 28 a + 22\right)\cdot 43 + \left(21 a^{3} + 8 a^{2} + 5 a + 17\right)\cdot 43^{2} + \left(9 a^{3} + 14 a^{2} + 21 a + 15\right)\cdot 43^{3} + \left(32 a^{3} + 9 a^{2} + 33 a + 41\right)\cdot 43^{4} + \left(9 a^{3} + 38 a^{2} + 12 a + 20\right)\cdot 43^{5} + \left(37 a^{3} + 22 a^{2} + 27 a + 18\right)\cdot 43^{6} + \left(27 a^{3} + 7 a^{2} + 5 a + 26\right)\cdot 43^{7} + \left(12 a^{3} + 41 a^{2} + 13 a + 22\right)\cdot 43^{8} + \left(31 a^{3} + 11 a + 4\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 5 }$ | $=$ | \( 40 a^{3} + 30 a^{2} + 27 a + 6 + \left(10 a^{3} + 41 a^{2} + 37 a + 14\right)\cdot 43 + \left(14 a^{3} + 4 a^{2} + 21 a + 40\right)\cdot 43^{2} + \left(34 a^{3} + 6 a^{2} + 23 a\right)\cdot 43^{3} + \left(38 a^{3} + 7 a^{2} + 3 a + 3\right)\cdot 43^{4} + \left(9 a^{3} + a^{2} + 15 a + 36\right)\cdot 43^{5} + \left(38 a^{3} + 38 a^{2} + 21 a + 36\right)\cdot 43^{6} + \left(6 a^{3} + 8 a^{2} + 14 a + 1\right)\cdot 43^{7} + \left(13 a^{3} + 36 a^{2} + 6 a + 36\right)\cdot 43^{8} + \left(11 a^{3} + 15 a^{2} + 27 a + 3\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 6 }$ | $=$ | \( 13 a^{3} + 19 a^{2} + 9 a + 7 + \left(19 a^{3} + 27 a^{2} + 38 a + 36\right)\cdot 43 + \left(3 a^{3} + 34 a^{2} + 39 a + 8\right)\cdot 43^{2} + \left(20 a^{3} + 24 a^{2} + 29 a + 13\right)\cdot 43^{3} + \left(8 a^{3} + 28 a^{2} + 38 a + 24\right)\cdot 43^{4} + \left(37 a^{3} + 26 a^{2} + 37 a + 17\right)\cdot 43^{5} + \left(3 a^{3} + 29 a^{2} + 13 a + 14\right)\cdot 43^{6} + \left(24 a^{3} + 12 a^{2} + 40 a + 38\right)\cdot 43^{7} + \left(38 a^{3} + 41 a^{2} + 42 a + 18\right)\cdot 43^{8} + \left(35 a^{3} + 22 a^{2} + 9 a + 37\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 7 }$ | $=$ | \( 12 a^{3} + 16 a^{2} + 42 a + 15 + \left(4 a^{3} + 2 a^{2} + 4 a + 22\right)\cdot 43 + \left(12 a^{3} + a^{2} + 32 a + 34\right)\cdot 43^{2} + \left(8 a^{3} + 33 a^{2} + 11 a + 2\right)\cdot 43^{3} + \left(21 a^{3} + 8 a^{2} + 15 a + 15\right)\cdot 43^{4} + \left(22 a^{3} + 3 a^{2} + 22 a + 12\right)\cdot 43^{5} + \left(20 a^{3} + 9 a^{2} + 5 a + 38\right)\cdot 43^{6} + \left(4 a^{3} + 2 a^{2} + 41 a + 28\right)\cdot 43^{7} + \left(27 a^{3} + 12 a^{2} + 34 a + 39\right)\cdot 43^{8} + \left(26 a^{3} + a^{2} + 29 a + 8\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 8 }$ | $=$ | \( 8 a^{3} + 24 a^{2} + 14 a + 15 + \left(6 a^{3} + 31 a^{2} + 26 a + 11\right)\cdot 43 + \left(32 a^{3} + a^{2} + 15 a + 15\right)\cdot 43^{2} + \left(35 a^{3} + 40 a^{2} + 12 a + 1\right)\cdot 43^{3} + \left(15 a^{3} + 33 a^{2} + 39 a + 13\right)\cdot 43^{4} + \left(17 a^{3} + a^{2} + 2 a + 36\right)\cdot 43^{5} + \left(41 a^{2} + 23 a + 13\right)\cdot 43^{6} + \left(8 a^{3} + 37 a^{2} + 32 a + 2\right)\cdot 43^{7} + \left(35 a^{3} + 34 a^{2} + 30 a + 17\right)\cdot 43^{8} + \left(28 a^{3} + 33 a^{2} + 19 a + 41\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 9 }$ | $=$ | \( 4 a^{3} + 5 a^{2} + 26 a + 11 + \left(10 a^{3} + 37 a^{2} + 35 a + 28\right)\cdot 43 + \left(a^{3} + 9 a^{2} + 13 a + 27\right)\cdot 43^{2} + \left(2 a^{3} + 21 a^{2} + 16 a + 37\right)\cdot 43^{3} + \left(29 a^{3} + 6 a^{2} + 20 a + 36\right)\cdot 43^{4} + \left(14 a^{3} + a^{2} + 39 a + 39\right)\cdot 43^{5} + \left(20 a^{3} + 9 a^{2} + 9 a + 8\right)\cdot 43^{6} + \left(27 a^{3} + 22 a^{2} + 34 a + 18\right)\cdot 43^{7} + \left(16 a^{3} + 25 a^{2} + 14 a + 41\right)\cdot 43^{8} + \left(42 a^{3} + 27 a^{2} + 29 a + 16\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 10 }$ | $=$ | \( 11 a^{3} + 41 a^{2} + 2 a + 3 + \left(27 a^{3} + 13 a^{2} + 23 a + 1\right)\cdot 43 + \left(33 a^{3} + 18 a^{2} + 19 a + 25\right)\cdot 43^{2} + \left(28 a^{3} + 17 a^{2} + 37 a + 7\right)\cdot 43^{3} + \left(9 a^{3} + 32 a^{2} + 3 a + 12\right)\cdot 43^{4} + \left(4 a^{3} + 12 a^{2} + 37 a + 41\right)\cdot 43^{5} + \left(39 a^{3} + 18 a^{2} + 42 a + 22\right)\cdot 43^{6} + \left(16 a^{3} + a^{2} + 18 a + 31\right)\cdot 43^{7} + \left(30 a^{3} + 29 a^{2} + 8 a + 13\right)\cdot 43^{8} + \left(11 a^{2} + 9 a + 35\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 11 }$ | $=$ | \( 27 a^{3} + 16 a^{2} + 38 a + 13 + \left(a^{3} + 33 a^{2} + 41 a + 1\right)\cdot 43 + \left(30 a^{3} + 3 a^{2} + 31 a + 40\right)\cdot 43^{2} + \left(2 a^{3} + 41 a^{2} + 14 a + 37\right)\cdot 43^{3} + \left(24 a^{3} + 21 a^{2} + 19 a + 5\right)\cdot 43^{4} + \left(29 a^{3} + 25 a^{2} + 2 a + 28\right)\cdot 43^{5} + \left(42 a^{3} + 15 a^{2} + 35 a + 13\right)\cdot 43^{6} + \left(9 a^{3} + 29 a^{2} + 7 a\right)\cdot 43^{7} + \left(39 a^{3} + 29 a^{2} + 2 a + 24\right)\cdot 43^{8} + \left(19 a^{3} + 30 a^{2} + 7 a + 32\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 12 }$ | $=$ | \( 21 a^{3} + 39 a^{2} + 15 a + 8 + \left(19 a^{3} + 36 a + 40\right)\cdot 43 + \left(41 a^{3} + 21 a^{2} + 13 a + 10\right)\cdot 43^{2} + \left(39 a^{3} + 39 a^{2} + 2 a + 31\right)\cdot 43^{3} + \left(24 a^{3} + 29 a^{2} + 23 a + 8\right)\cdot 43^{4} + \left(14 a^{3} + 33 a^{2} + 32\right)\cdot 43^{5} + \left(7 a^{3} + 18 a^{2} + 14 a + 10\right)\cdot 43^{6} + \left(23 a^{3} + 37 a^{2} + 11 a + 28\right)\cdot 43^{7} + \left(33 a^{2} + 21 a + 6\right)\cdot 43^{8} + \left(41 a^{3} + 21 a^{2} + 22 a + 2\right)\cdot 43^{9} +O(43^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character value |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-3$ |
$3$ | $2$ | $(1,7)(4,10)$ | $1$ |
$3$ | $2$ | $(2,8)(3,9)(5,11)(6,12)$ | $-1$ |
$4$ | $3$ | $(1,8,6)(2,12,7)(3,10,5)(4,11,9)$ | $0$ |
$4$ | $3$ | $(1,6,8)(2,7,12)(3,5,10)(4,9,11)$ | $0$ |
$1$ | $4$ | $(1,4,7,10)(2,5,8,11)(3,6,9,12)$ | $3 \zeta_{4}$ |
$1$ | $4$ | $(1,10,7,4)(2,11,8,5)(3,12,9,6)$ | $-3 \zeta_{4}$ |
$3$ | $4$ | $(1,10,7,4)(2,5,8,11)(3,6,9,12)$ | $\zeta_{4}$ |
$3$ | $4$ | $(1,4,7,10)(2,11,8,5)(3,12,9,6)$ | $-\zeta_{4}$ |
$4$ | $6$ | $(1,12,8,7,6,2)(3,11,10,9,5,4)$ | $0$ |
$4$ | $6$ | $(1,2,6,7,8,12)(3,4,5,9,10,11)$ | $0$ |
$4$ | $12$ | $(1,5,12,4,8,3,7,11,6,10,2,9)$ | $0$ |
$4$ | $12$ | $(1,3,2,4,6,5,7,9,8,10,12,11)$ | $0$ |
$4$ | $12$ | $(1,11,12,10,8,9,7,5,6,4,2,3)$ | $0$ |
$4$ | $12$ | $(1,9,2,10,6,11,7,3,8,4,12,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.