Basic invariants
Dimension: | $11$ |
Group: | $\PSL(2,11)$ |
Conductor: | \(284\!\cdots\!921\)\(\medspace = 19^{6} \cdot 3659^{6} \cdot 3688801^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 11.3.4325187603056652501797342844511965821771743681.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PSL(2,11)$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $\PSL(2,11)$ |
Projective stem field: | Galois closure of 11.3.4325187603056652501797342844511965821771743681.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{11} - 33 x^{9} - 176 x^{8} - 1881 x^{7} - 9768 x^{6} - 96962 x^{5} - 970164 x^{4} - 4535019 x^{3} + \cdots - 3808296 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{5} + 8x + 40 \)
Roots:
$r_{ 1 }$ | $=$ | \( 18 + 31\cdot 43 + 35\cdot 43^{2} + 16\cdot 43^{3} + 19\cdot 43^{4} + 14\cdot 43^{5} + 17\cdot 43^{6} + 24\cdot 43^{7} + 17\cdot 43^{8} + 28\cdot 43^{9} +O(43^{10})\) |
$r_{ 2 }$ | $=$ | \( 5 a^{4} + 8 a^{3} + 13 a^{2} + 42 a + 25 + \left(29 a^{4} + 10 a^{3} + 24 a^{2} + 34 a + 15\right)\cdot 43 + \left(30 a^{4} + 18 a^{3} + 7 a^{2} + 8 a + 22\right)\cdot 43^{2} + \left(a^{4} + 21 a^{3} + 35 a^{2} + 3 a + 42\right)\cdot 43^{3} + \left(4 a^{4} + 16 a^{3} + 21 a^{2} + 41 a + 42\right)\cdot 43^{4} + \left(38 a^{4} + 40 a^{3} + 3 a^{2} + 35 a + 21\right)\cdot 43^{5} + \left(21 a^{4} + 26 a^{3} + 28 a^{2} + 32 a + 38\right)\cdot 43^{6} + \left(40 a^{4} + 20 a^{3} + 27 a^{2} + 41 a + 2\right)\cdot 43^{7} + \left(26 a^{4} + 3 a^{3} + 37 a^{2} + 34 a + 41\right)\cdot 43^{8} + \left(21 a^{4} + 29 a^{3} + 34 a^{2} + 27 a + 8\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 3 }$ | $=$ | \( 10 a^{4} + 2 a^{3} + 39 a^{2} + 30 a + 33 + \left(16 a^{4} + 13 a^{3} + 30 a^{2} + 32 a + 1\right)\cdot 43 + \left(a^{4} + 18 a^{3} + 15 a + 38\right)\cdot 43^{2} + \left(26 a^{4} + 19 a^{3} + a^{2} + 40 a + 19\right)\cdot 43^{3} + \left(20 a^{4} + 39 a^{3} + 30 a^{2} + 10 a + 33\right)\cdot 43^{4} + \left(13 a^{4} + 4 a^{3} + 22 a^{2} + 26 a + 12\right)\cdot 43^{5} + \left(41 a^{4} + 5 a^{3} + 32 a^{2} + 42 a + 1\right)\cdot 43^{6} + \left(9 a^{4} + 31 a^{3} + 22 a^{2} + 36 a + 40\right)\cdot 43^{7} + \left(8 a^{4} + a^{3} + 31 a^{2} + 29 a + 42\right)\cdot 43^{8} + \left(27 a^{4} + 11 a^{3} + 27 a^{2} + 13 a + 13\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 4 }$ | $=$ | \( 13 a^{4} + 32 a^{3} + 2 a^{2} + 40 a + 16 + \left(37 a^{4} + 23 a^{3} + 30 a^{2} + 9 a + 16\right)\cdot 43 + \left(2 a^{4} + 21 a^{3} + 7 a^{2} + 32 a + 16\right)\cdot 43^{2} + \left(12 a^{4} + 5 a^{3} + 3 a^{2} + 39 a + 31\right)\cdot 43^{3} + \left(7 a^{4} + 7 a^{3} + 41 a^{2} + 25 a + 20\right)\cdot 43^{4} + \left(16 a^{4} + 12 a^{3} + 12 a^{2} + 9 a + 36\right)\cdot 43^{5} + \left(39 a^{3} + 37 a^{2} + 4 a + 29\right)\cdot 43^{6} + \left(29 a^{4} + 16 a^{3} + 14 a^{2} + 32 a + 6\right)\cdot 43^{7} + \left(31 a^{4} + 26 a^{3} + 7 a^{2} + 12 a + 11\right)\cdot 43^{8} + \left(7 a^{4} + 36 a^{3} + 28 a^{2} + 5 a + 23\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 5 }$ | $=$ | \( 14 a^{4} + 36 a^{3} + 40 a^{2} + 11 a + 31 + \left(40 a^{4} + 26 a^{3} + 6 a^{2} + 10 a + 35\right)\cdot 43 + \left(42 a^{4} + 32 a^{3} + 41 a^{2} + 30 a + 14\right)\cdot 43^{2} + \left(36 a^{4} + 9 a^{3} + 33 a^{2} + 41 a + 10\right)\cdot 43^{3} + \left(36 a^{4} + 15 a^{3} + 35 a^{2} + 27 a + 12\right)\cdot 43^{4} + \left(16 a^{4} + 6 a^{2} + 10 a + 32\right)\cdot 43^{5} + \left(28 a^{4} + 38 a^{3} + 10 a^{2} + 17 a + 2\right)\cdot 43^{6} + \left(21 a^{4} + 22 a^{3} + 22 a^{2} + 27 a + 37\right)\cdot 43^{7} + \left(16 a^{4} + 17 a^{3} + 28 a^{2} + 2 a + 25\right)\cdot 43^{8} + \left(23 a^{4} + 30 a^{3} + 34 a^{2} + 38 a + 11\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 6 }$ | $=$ | \( 22 a^{4} + 15 a^{3} + 21 a^{2} + 30 a + 22 + \left(35 a^{4} + 20 a^{3} + 22 a^{2} + 13 a + 13\right)\cdot 43 + \left(15 a^{4} + 41 a^{3} + 36 a^{2} + 6 a + 13\right)\cdot 43^{2} + \left(41 a^{4} + 32 a^{3} + 26 a^{2} + 33 a + 38\right)\cdot 43^{3} + \left(4 a^{4} + 17 a^{3} + 32 a^{2} + 16 a + 5\right)\cdot 43^{4} + \left(34 a^{4} + 26 a^{3} + 9 a^{2} + 34 a + 5\right)\cdot 43^{5} + \left(28 a^{4} + 3 a^{3} + 27 a^{2} + 31\right)\cdot 43^{6} + \left(10 a^{4} + 31 a^{3} + 40 a^{2} + 14 a + 9\right)\cdot 43^{7} + \left(24 a^{4} + 30 a^{3} + 35 a^{2} + 42 a + 15\right)\cdot 43^{8} + \left(2 a^{4} + 33 a^{3} + 25 a^{2} + 12 a + 7\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 7 }$ | $=$ | \( 23 a^{4} + 15 a^{3} + 31 a^{2} + 7 a + 13 + \left(5 a^{4} + 23 a^{3} + 27 a^{2} + 23 a + 19\right)\cdot 43 + \left(30 a^{4} + 19 a^{3} + 14 a^{2} + 32 a + 41\right)\cdot 43^{2} + \left(15 a^{4} + 22 a^{3} + 28 a^{2} + 39 a + 13\right)\cdot 43^{3} + \left(33 a^{4} + 20 a^{3} + 34 a^{2} + 42 a + 3\right)\cdot 43^{4} + \left(35 a^{4} + a^{3} + 21 a^{2} + a + 35\right)\cdot 43^{5} + \left(17 a^{4} + 35 a^{3} + 4 a^{2} + 16 a + 5\right)\cdot 43^{6} + \left(39 a^{4} + 10 a^{3} + 32 a^{2} + 32 a + 22\right)\cdot 43^{7} + \left(34 a^{4} + 17 a^{3} + 2 a^{2} + 37 a + 24\right)\cdot 43^{8} + \left(2 a^{4} + 42 a^{3} + 4 a^{2} + 32 a + 38\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 8 }$ | $=$ | \( 27 a^{4} + 29 a^{3} + 11 a^{2} + 41 a + 30 + \left(39 a^{4} + 26 a^{3} + 22 a^{2} + 42 a + 39\right)\cdot 43 + \left(9 a^{4} + 27 a^{3} + 40 a^{2} + 11 a + 23\right)\cdot 43^{2} + \left(26 a^{4} + 28 a^{3} + 33 a^{2} + 30 a + 12\right)\cdot 43^{3} + \left(29 a^{4} + 5 a^{3} + 26 a^{2} + 11 a + 22\right)\cdot 43^{4} + \left(29 a^{4} + 23 a^{3} + 37 a^{2} + 16 a + 30\right)\cdot 43^{5} + \left(17 a^{4} + 40 a^{3} + 39 a + 30\right)\cdot 43^{6} + \left(5 a^{4} + 10 a^{3} + 9 a^{2} + 16 a + 36\right)\cdot 43^{7} + \left(27 a^{4} + 12 a^{3} + 8 a^{2} + 9 a + 34\right)\cdot 43^{8} + \left(27 a^{4} + 33 a^{3} + 37 a^{2} + a + 16\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 9 }$ | $=$ | \( 31 a^{4} + 26 a^{3} + 13 a^{2} + 19 a + 4 + \left(41 a^{4} + 42 a^{3} + 10 a^{2} + 37 a + 10\right)\cdot 43 + \left(32 a^{4} + 34 a^{3} + 28 a^{2} + 20 a + 25\right)\cdot 43^{2} + \left(36 a^{4} + 5 a^{3} + 28 a^{2} + 39 a + 2\right)\cdot 43^{3} + \left(14 a^{4} + 10 a^{3} + 26 a^{2} + 28 a + 31\right)\cdot 43^{4} + \left(28 a^{4} + 3 a^{3} + 19 a^{2} + 41 a + 4\right)\cdot 43^{5} + \left(a^{4} + 13 a^{3} + 25 a^{2} + 29 a + 14\right)\cdot 43^{6} + \left(26 a^{4} + 4 a^{3} + 20 a^{2} + 22 a + 5\right)\cdot 43^{7} + \left(28 a^{4} + 19 a^{3} + 39 a^{2} + 38 a + 27\right)\cdot 43^{8} + \left(a^{4} + 23 a^{3} + 30 a^{2} + 9 a + 22\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 10 }$ | $=$ | \( 32 a^{4} + 38 a^{3} + 10 a^{2} + 6 a + \left(29 a^{4} + 4 a^{3} + 2 a^{2} + 17 a + 11\right)\cdot 43 + \left(36 a^{4} + 15 a^{3} + 36 a^{2} + 8 a + 9\right)\cdot 43^{2} + \left(36 a^{4} + 16 a^{3} + 29 a^{2} + 11 a + 35\right)\cdot 43^{3} + \left(32 a^{4} + 29 a^{3} + 40 a^{2} + 17 a + 3\right)\cdot 43^{4} + \left(23 a^{4} + 6 a^{3} + 9 a^{2} + 38 a + 42\right)\cdot 43^{5} + \left(6 a^{4} + 21 a^{3} + 26 a^{2} + 30 a + 34\right)\cdot 43^{6} + \left(27 a^{4} + 37 a^{3} + 23 a^{2} + 13 a + 37\right)\cdot 43^{7} + \left(29 a^{4} + 7 a^{3} + 19 a^{2} + 36 a + 23\right)\cdot 43^{8} + \left(30 a^{4} + 42 a^{3} + 5 a^{2} + a + 32\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 11 }$ | $=$ | \( 38 a^{4} + 14 a^{3} + 35 a^{2} + 32 a + 23 + \left(25 a^{4} + 23 a^{3} + 37 a^{2} + 35 a + 20\right)\cdot 43 + \left(11 a^{4} + 28 a^{3} + a^{2} + 4 a + 17\right)\cdot 43^{2} + \left(24 a^{4} + 9 a^{3} + 37 a^{2} + 22 a + 34\right)\cdot 43^{3} + \left(30 a^{4} + 10 a^{3} + 10 a^{2} + 34 a + 19\right)\cdot 43^{4} + \left(21 a^{4} + 10 a^{3} + 27 a^{2} + 42 a + 22\right)\cdot 43^{5} + \left(7 a^{4} + 35 a^{3} + 22 a^{2} + 8\right)\cdot 43^{6} + \left(5 a^{4} + 28 a^{3} + a^{2} + 20 a + 35\right)\cdot 43^{7} + \left(30 a^{4} + 35 a^{3} + 4 a^{2} + 13 a + 36\right)\cdot 43^{8} + \left(26 a^{4} + 18 a^{3} + 29 a^{2} + 28 a + 10\right)\cdot 43^{9} +O(43^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 11 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 11 }$ | Character value |
$1$ | $1$ | $()$ | $11$ |
$55$ | $2$ | $(1,9)(2,7)(3,10)(6,11)$ | $-1$ |
$110$ | $3$ | $(1,8,11)(3,9,7)(4,10,5)$ | $-1$ |
$132$ | $5$ | $(1,2,4,7,9)(3,10,11,8,6)$ | $1$ |
$132$ | $5$ | $(1,4,9,2,7)(3,11,6,10,8)$ | $1$ |
$110$ | $6$ | $(1,8,11,7,6,10)(2,4,3)(5,9)$ | $-1$ |
$60$ | $11$ | $(1,7,2,3,5,4,10,9,8,11,6)$ | $0$ |
$60$ | $11$ | $(1,2,5,10,8,6,7,3,4,9,11)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.