Basic invariants
Dimension: | $3$ |
Group: | 12T29 |
Conductor: | \(107653\)\(\medspace = 7^{2} \cdot 13^{3} \) |
Artin stem field: | Galois closure of 12.8.61132828589969773.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 12T29 |
Parity: | odd |
Determinant: | 1.13.4t1.a.b |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.8281.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - 3 x^{11} - 9 x^{10} + 29 x^{9} + 41 x^{8} - 82 x^{7} - 263 x^{6} + 36 x^{5} + 705 x^{4} + \cdots - 13 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{4} + 23x + 6 \)
Roots:
$r_{ 1 }$ | $=$ | \( 23 a^{3} + 24 a^{2} + 20 a + 3 + \left(36 a^{3} + 36 a^{2} + 13 a + 25\right)\cdot 41 + \left(38 a^{3} + 17 a^{2} + 40 a + 29\right)\cdot 41^{2} + \left(37 a^{3} + 32 a^{2} + 7 a + 35\right)\cdot 41^{3} + \left(24 a^{3} + 3 a^{2} + 32 a + 4\right)\cdot 41^{4} + \left(9 a^{3} + 27 a^{2} + 16 a + 18\right)\cdot 41^{5} + \left(7 a^{3} + 23 a^{2} + 18 a + 8\right)\cdot 41^{6} + \left(4 a^{3} + 32 a^{2} + 15 a + 5\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 2 }$ | $=$ | \( 7 a^{3} + 13 a^{2} + 6 a + 27 + \left(a^{3} + 37 a^{2} + 13 a + 19\right)\cdot 41 + \left(27 a^{3} + 17 a^{2} + 29 a + 39\right)\cdot 41^{2} + \left(15 a^{3} + 32 a^{2} + 16 a + 21\right)\cdot 41^{3} + \left(19 a^{3} + 3 a^{2} + 24 a + 24\right)\cdot 41^{4} + \left(26 a^{3} + 5 a^{2} + a + 12\right)\cdot 41^{5} + \left(a^{3} + 26 a^{2} + 11 a + 17\right)\cdot 41^{6} + \left(26 a^{3} + 13 a^{2} + 22 a + 13\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 3 }$ | $=$ | \( 15 a^{3} + 39 a^{2} + 28 a + 29 + \left(13 a^{3} + 22 a^{2} + 39 a + 24\right)\cdot 41 + \left(5 a^{3} + 7 a^{2} + 11 a + 24\right)\cdot 41^{2} + \left(12 a^{3} + 24 a^{2} + 18 a + 10\right)\cdot 41^{3} + \left(25 a^{3} + 29 a^{2} + 27 a + 11\right)\cdot 41^{4} + \left(20 a^{3} + 13 a^{2} + 40 a + 13\right)\cdot 41^{5} + \left(37 a^{3} + 2 a^{2} + 27 a + 28\right)\cdot 41^{6} + \left(16 a^{3} + 4 a^{2} + 9\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 4 }$ | $=$ | \( 39 a^{3} + 20 a^{2} + 20 a + 13 + \left(39 a^{3} + 31 a^{2} + 5 a + 24\right)\cdot 41 + \left(22 a^{3} + 11 a^{2} + 40 a + 39\right)\cdot 41^{2} + \left(7 a^{3} + 27 a + 33\right)\cdot 41^{3} + \left(36 a^{3} + 34 a^{2} + 38 a + 36\right)\cdot 41^{4} + \left(33 a^{3} + 33 a^{2} + 20 a + 17\right)\cdot 41^{5} + \left(13 a^{3} + 24 a^{2} + 11 a + 17\right)\cdot 41^{6} + \left(24 a^{3} + 6 a^{2} + 22 a + 20\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 5 }$ | $=$ | \( 21 a^{3} + 2 a^{2} + 20 a + 10 + \left(20 a^{3} + 7 a^{2} + 8 a + 27\right)\cdot 41 + \left(6 a^{3} + 29 a^{2} + 18 a + 11\right)\cdot 41^{2} + \left(24 a^{3} + 9 a^{2} + 17 a + 33\right)\cdot 41^{3} + \left(23 a^{2} + 2 a + 37\right)\cdot 41^{4} + \left(23 a^{3} + 18 a^{2} + 23 a + 14\right)\cdot 41^{5} + \left(25 a^{3} + 15 a^{2} + 32 a + 35\right)\cdot 41^{6} + \left(5 a^{3} + 37 a^{2} + 39 a + 35\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 6 }$ | $=$ | \( 4 a^{3} + 39 a^{2} + 6 a + 6 + \left(39 a^{3} + 6 a^{2} + 16 a + 28\right)\cdot 41 + \left(4 a^{3} + 27 a^{2} + 22 a + 27\right)\cdot 41^{2} + \left(11 a^{3} + 4 a^{2} + 5 a + 25\right)\cdot 41^{3} + \left(a^{3} + 7 a^{2} + 8 a + 19\right)\cdot 41^{4} + \left(31 a^{3} + 40 a^{2} + 13 a + 19\right)\cdot 41^{5} + \left(40 a^{3} + 6 a^{2} + 9 a + 15\right)\cdot 41^{6} + \left(32 a^{3} + 8 a^{2} + 3 a + 10\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 7 }$ | $=$ | \( 33 a^{3} + 13 a^{2} + 19 a + 32 + \left(37 a^{3} + 38 a^{2} + 24 a + 15\right)\cdot 41 + \left(16 a^{3} + 12 a^{2} + 21 a + 9\right)\cdot 41^{2} + \left(30 a^{3} + 11 a^{2} + 32 a + 8\right)\cdot 41^{3} + \left(29 a^{3} + 6 a^{2} + 38 a + 16\right)\cdot 41^{4} + \left(10 a^{3} + 8 a^{2} + a + 37\right)\cdot 41^{5} + \left(27 a^{3} + 11 a^{2} + 10 a + 25\right)\cdot 41^{6} + \left(30 a^{3} + 8 a^{2} + 30 a + 31\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 8 }$ | $=$ | \( 27 a^{3} + 8 a^{2} + 25 a + 3 + \left(6 a^{3} + 35 a^{2} + 25 a + 22\right)\cdot 41 + \left(2 a^{3} + 14 a^{2} + 21 a + 20\right)\cdot 41^{2} + \left(32 a^{3} + 30 a^{2} + 13 a + 7\right)\cdot 41^{3} + \left(20 a^{3} + 27 a^{2} + 22 a + 28\right)\cdot 41^{4} + \left(35 a^{3} + 24 a^{2} + 2 a + 14\right)\cdot 41^{5} + \left(23 a^{3} + 18 a^{2} + 29 a + 21\right)\cdot 41^{6} + \left(30 a^{3} + 40 a^{2} + a + 40\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 9 }$ | $=$ | \( 11 a^{3} + 6 a^{2} + 15 a + 1 + \left(35 a^{3} + 25 a^{2} + 4 a + 13\right)\cdot 41 + \left(20 a^{3} + 2 a^{2} + 8 a + 36\right)\cdot 41^{2} + \left(a^{3} + 14 a^{2} + 23 a + 11\right)\cdot 41^{3} + \left(2 a^{3} + a^{2} + 24 a + 30\right)\cdot 41^{4} + \left(33 a^{2} + 22 a + 37\right)\cdot 41^{5} + \left(10 a^{3} + 3 a^{2} + 25 a + 4\right)\cdot 41^{6} + \left(30 a^{3} + 37 a^{2} + 35 a + 14\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 10 }$ | $=$ | \( 4 a^{3} + 15 a^{2} + 18 a + 14 + \left(19 a^{3} + 33 a^{2} + 17 a + 23\right)\cdot 41 + \left(29 a^{3} + 7 a^{2} + 27 a + 28\right)\cdot 41^{2} + \left(34 a^{3} + 37 a^{2} + 18 a + 20\right)\cdot 41^{3} + \left(12 a^{3} + 18 a^{2} + 20 a + 13\right)\cdot 41^{4} + \left(27 a^{3} + 5 a^{2} + 17 a + 17\right)\cdot 41^{5} + \left(20 a^{3} + 35 a^{2} + 28 a + 12\right)\cdot 41^{6} + \left(32 a^{3} + 2 a^{2} + 34 a + 38\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 11 }$ | $=$ | \( 18 a^{3} + 4 a^{2} + 24 a + 30 + \left(2 a^{3} + 10 a^{2} + 9 a + 12\right)\cdot 41 + \left(23 a^{3} + 33 a^{2} + 37 a + 10\right)\cdot 41^{2} + \left(15 a^{3} + 34 a^{2} + 17 a + 18\right)\cdot 41^{3} + \left(32 a^{3} + 5 a^{2} + 20 a + 22\right)\cdot 41^{4} + \left(38 a^{3} + 24 a^{2} + 20 a + 20\right)\cdot 41^{5} + \left(21 a^{3} + 6 a^{2} + 9 a + 34\right)\cdot 41^{6} + \left(19 a^{3} + 35 a^{2} + 26 a + 29\right)\cdot 41^{7} +O(41^{8})\) |
$r_{ 12 }$ | $=$ | \( 3 a^{3} + 22 a^{2} + 4 a + 40 + \left(35 a^{3} + 2 a^{2} + 27 a + 9\right)\cdot 41 + \left(6 a^{3} + 22 a^{2} + 8 a + 9\right)\cdot 41^{2} + \left(23 a^{3} + 14 a^{2} + 5 a + 18\right)\cdot 41^{3} + \left(40 a^{3} + 2 a^{2} + 27 a\right)\cdot 41^{4} + \left(29 a^{3} + 12 a^{2} + 23 a + 22\right)\cdot 41^{5} + \left(15 a^{3} + 30 a^{2} + 32 a + 24\right)\cdot 41^{6} + \left(33 a^{3} + 19 a^{2} + 13 a + 37\right)\cdot 41^{7} +O(41^{8})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ | $-3$ |
$3$ | $2$ | $(1,7)(3,9)$ | $1$ |
$3$ | $2$ | $(2,8)(4,10)(5,11)(6,12)$ | $-1$ |
$4$ | $3$ | $(1,11,12)(2,3,4)(5,6,7)(8,9,10)$ | $0$ |
$4$ | $3$ | $(1,12,11)(2,4,3)(5,7,6)(8,10,9)$ | $0$ |
$1$ | $4$ | $(1,3,7,9)(2,6,8,12)(4,5,10,11)$ | $-3 \zeta_{4}$ |
$1$ | $4$ | $(1,9,7,3)(2,12,8,6)(4,11,10,5)$ | $3 \zeta_{4}$ |
$3$ | $4$ | $(1,9,7,3)(2,6,8,12)(4,5,10,11)$ | $-\zeta_{4}$ |
$3$ | $4$ | $(1,3,7,9)(2,12,8,6)(4,11,10,5)$ | $\zeta_{4}$ |
$4$ | $6$ | $(1,6,11,7,12,5)(2,10,3,8,4,9)$ | $0$ |
$4$ | $6$ | $(1,5,12,7,11,6)(2,9,4,8,3,10)$ | $0$ |
$4$ | $12$ | $(1,10,6,3,11,8,7,4,12,9,5,2)$ | $0$ |
$4$ | $12$ | $(1,8,5,3,12,10,7,2,11,9,6,4)$ | $0$ |
$4$ | $12$ | $(1,4,6,9,11,2,7,10,12,3,5,8)$ | $0$ |
$4$ | $12$ | $(1,2,5,9,12,4,7,8,11,3,6,10)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.