Basic invariants
Dimension: | $11$ |
Group: | $\PGL(2,11)$ |
Conductor: | \(750\!\cdots\!576\)\(\medspace = 2^{10} \cdot 7^{10} \cdot 11^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 12.2.82527728843210964110336.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 24T2949 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $\PGL(2,11)$ |
Projective stem field: | Galois closure of 12.2.82527728843210964110336.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{12} - x^{11} - 11x^{10} - 55x^{9} - 66x^{7} - 154x^{6} + 66x^{5} + 165x^{4} + 275x^{3} - 11x^{2} + 21x - 758 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{6} + 19x^{3} + 16x^{2} + 8x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 12 a^{5} + 19 a^{4} + 23 a^{3} + 8 a^{2} + 12 a + 15 + \left(13 a^{5} + 21 a^{3} + a^{2} + 22 a + 2\right)\cdot 31 + \left(27 a^{5} + 10 a^{4} + 2 a^{3} + 14 a^{2} + 22 a + 9\right)\cdot 31^{2} + \left(26 a^{5} + 28 a^{4} + 12 a^{3} + 24 a^{2} + 22 a + 18\right)\cdot 31^{3} + \left(24 a^{5} + 2 a^{4} + 29 a^{3} + 29 a^{2} + 28 a + 25\right)\cdot 31^{4} + \left(18 a^{5} + 28 a^{4} + 19 a^{3} + 27 a^{2} + 19 a + 25\right)\cdot 31^{5} + \left(16 a^{5} + 6 a^{4} + 9 a^{3} + 4 a^{2} + 21 a + 7\right)\cdot 31^{6} + \left(30 a^{5} + 23 a^{4} + 6 a^{3} + 23 a^{2} + 13 a + 10\right)\cdot 31^{7} + \left(11 a^{5} + 6 a^{4} + 21 a^{3} + 15 a^{2} + 16 a + 29\right)\cdot 31^{8} + \left(19 a^{5} + 28 a^{4} + 6 a^{3} + 5 a^{2} + 30 a + 27\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 9 a^{5} + 5 a^{4} + 3 a^{3} + 9 a^{2} + 5 a + 7 + \left(30 a^{5} + 18 a^{4} + 27 a^{3} + 3 a^{2} + 12 a + 12\right)\cdot 31 + \left(26 a^{5} + 22 a^{4} + 13 a^{3} + 2 a^{2} + 23 a + 3\right)\cdot 31^{2} + \left(a^{5} + 11 a^{4} + 15 a^{3} + 5 a^{2} + 27 a + 6\right)\cdot 31^{3} + \left(25 a^{5} + 17 a^{4} + 12 a^{3} + 13 a^{2} + 10 a + 5\right)\cdot 31^{4} + \left(a^{5} + 16 a^{4} + 16 a^{3} + 11 a^{2} + 28 a + 14\right)\cdot 31^{5} + \left(13 a^{5} + 17 a^{4} + 23 a^{3} + 21 a + 12\right)\cdot 31^{6} + \left(26 a^{5} + 20 a^{4} + 27 a^{2} + 3 a + 15\right)\cdot 31^{7} + \left(10 a^{5} + 19 a^{4} + 9 a^{3} + 20 a^{2} + 16 a + 2\right)\cdot 31^{8} + \left(13 a^{5} + 20 a^{4} + 17 a^{3} + 6 a^{2} + 5 a + 23\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 12 a^{5} + 5 a^{4} + 30 a^{3} + 20 a^{2} + 30 a + 16 + \left(2 a^{5} + 29 a^{4} + 12 a^{3} + 8 a + 21\right)\cdot 31 + \left(28 a^{5} + 16 a^{4} + 18 a^{3} + 21 a^{2} + 13 a + 27\right)\cdot 31^{2} + \left(18 a^{5} + 27 a^{4} + 13 a^{3} + 2 a^{2} + 25 a + 10\right)\cdot 31^{3} + \left(13 a^{5} + 2 a^{4} + 7 a^{3} + 29 a + 12\right)\cdot 31^{4} + \left(25 a^{5} + 2 a^{4} + 28 a^{3} + 10 a^{2} + 3 a + 10\right)\cdot 31^{5} + \left(18 a^{5} + 10 a^{4} + 6 a^{3} + 22 a^{2} + 12 a + 29\right)\cdot 31^{6} + \left(23 a^{5} + 7 a^{4} + 19 a^{3} + 25 a^{2} + 29 a + 14\right)\cdot 31^{7} + \left(9 a^{5} + 19 a^{4} + 13 a^{3} + 24 a^{2} + 23 a + 5\right)\cdot 31^{8} + \left(12 a^{5} + 30 a^{4} + 12 a^{3} + 9 a^{2} + 16\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( a^{5} + 28 a^{4} + 11 a^{3} + 10 a^{2} + 10 a + 27 + \left(14 a^{5} + 28 a^{4} + 6 a^{3} + 8 a^{2} + 22 a + 22\right)\cdot 31 + \left(26 a^{5} + 11 a^{4} + 5 a^{3} + 29 a^{2} + 16 a + 15\right)\cdot 31^{2} + \left(19 a^{5} + 26 a^{4} + 26 a^{3} + 22 a^{2} + 26 a + 22\right)\cdot 31^{3} + \left(22 a^{5} + 10 a^{4} + a^{3} + 12 a^{2} + 12 a + 19\right)\cdot 31^{4} + \left(18 a^{5} + 3 a^{4} + 27 a^{3} + 10 a^{2} + 9 a + 4\right)\cdot 31^{5} + \left(24 a^{5} + 18 a^{4} + 29 a^{3} + 9 a^{2} + 10 a + 11\right)\cdot 31^{6} + \left(19 a^{4} + 30 a^{3} + 26 a^{2} + 15 a + 7\right)\cdot 31^{7} + \left(10 a^{5} + 14 a^{4} + 17 a^{3} + 11 a^{2} + 28 a + 18\right)\cdot 31^{8} + \left(6 a^{5} + 21 a^{4} + 26 a^{3} + 6 a^{2} + 28 a + 16\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 22 a^{5} + 11 a^{4} + 29 a^{3} + 5 a^{2} + 7 a + 8 + \left(20 a^{4} + 30 a^{3} + 16 a^{2} + 28 a + 30\right)\cdot 31 + \left(9 a^{5} + 30 a^{4} + 15 a^{3} + 17 a^{2} + 23\right)\cdot 31^{2} + \left(23 a^{4} + 25 a^{3} + 3 a^{2} + 8 a + 13\right)\cdot 31^{3} + \left(29 a^{5} + 3 a^{4} + 3 a^{3} + 2 a^{2} + 20 a + 7\right)\cdot 31^{4} + \left(3 a^{5} + 13 a^{4} + 3 a^{3} + 8 a^{2} + 24 a + 30\right)\cdot 31^{5} + \left(8 a^{5} + 16 a^{4} + 16 a^{3} + 30 a^{2} + 14\right)\cdot 31^{6} + \left(30 a^{5} + 2 a^{4} + 29 a^{3} + 12 a^{2} + 28\right)\cdot 31^{7} + \left(5 a^{5} + 28 a^{4} + 25 a^{3} + 16 a^{2} + 9 a + 10\right)\cdot 31^{8} + \left(20 a^{5} + 4 a^{4} + 15 a^{3} + 29 a^{2} + 16 a + 2\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 24 a^{5} + 7 a^{4} + 30 a^{2} + 26 a + 8 + \left(14 a^{5} + 24 a^{4} + 13 a^{3} + 6 a^{2} + 21 a + 5\right)\cdot 31 + \left(21 a^{5} + 16 a^{4} + 29 a^{3} + 16 a^{2} + 17 a + 14\right)\cdot 31^{2} + \left(9 a^{5} + 9 a^{4} + 9 a^{3} + 15 a^{2} + 12 a + 22\right)\cdot 31^{3} + \left(20 a^{5} + 22 a^{4} + 17 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 31^{4} + \left(19 a^{5} + 8 a^{4} + 15 a^{3} + 15 a^{2} + 12 a + 20\right)\cdot 31^{5} + \left(26 a^{5} + 6 a^{4} + 13 a^{3} + 21 a^{2} + 12 a + 10\right)\cdot 31^{6} + \left(12 a^{5} + 17 a^{4} + 28 a^{3} + 7 a^{2} + 8 a + 17\right)\cdot 31^{7} + \left(12 a^{5} + 3 a^{4} + 3 a^{3} + 9 a^{2} + 6\right)\cdot 31^{8} + \left(19 a^{5} + 12 a^{4} + 28 a^{3} + 21 a^{2} + 2 a + 20\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 13 a^{5} + 15 a^{4} + 19 a^{3} + 15 a^{2} + 15 a + 3 + \left(26 a^{5} + 7 a^{4} + 17 a^{3} + 8 a^{2} + 13 a + 20\right)\cdot 31 + \left(8 a^{5} + 4 a^{4} + 2 a^{3} + 26 a^{2} + 4 a + 18\right)\cdot 31^{2} + \left(30 a^{5} + 16 a^{4} + 5 a^{3} + 29 a^{2} + 18 a + 4\right)\cdot 31^{3} + \left(26 a^{5} + 29 a^{4} + 20 a^{3} + a^{2} + 26 a + 29\right)\cdot 31^{4} + \left(7 a^{5} + 16 a^{4} + 12 a^{3} + 18 a^{2} + 3 a + 17\right)\cdot 31^{5} + \left(15 a^{5} + 26 a^{4} + 14 a^{3} + 4 a + 5\right)\cdot 31^{6} + \left(11 a^{5} + 4 a^{4} + 23 a^{3} + 20 a^{2} + 20 a\right)\cdot 31^{7} + \left(30 a^{5} + 12 a^{4} + 28 a^{3} + 3 a^{2} + 19 a + 23\right)\cdot 31^{8} + \left(2 a^{5} + 18 a^{4} + 20 a^{3} + 16 a^{2} + 15 a\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 27 a^{5} + 8 a^{4} + 24 a^{3} + 8 a^{2} + 19 a + 2 + \left(11 a^{5} + 4 a^{4} + 22 a^{3} + 8 a^{2} + a + 10\right)\cdot 31 + \left(24 a^{5} + 11 a^{4} + 5 a^{3} + 5 a^{2} + a + 24\right)\cdot 31^{2} + \left(25 a^{5} + 25 a^{4} + 16 a^{3} + 19 a^{2} + 2 a + 2\right)\cdot 31^{3} + \left(12 a^{5} + 14 a^{4} + 5 a^{3} + 6 a^{2} + 25 a + 16\right)\cdot 31^{4} + \left(25 a^{5} + 13 a^{4} + 9 a^{3} + 12 a^{2} + 14 a + 23\right)\cdot 31^{5} + \left(25 a^{5} + 25 a^{4} + 21 a^{3} + a^{2} + 21 a + 10\right)\cdot 31^{6} + \left(4 a^{5} + 15 a^{4} + 10 a^{3} + 18 a^{2} + 7 a + 13\right)\cdot 31^{7} + \left(9 a^{5} + 29 a^{4} + 24 a^{3} + 2 a^{2} + 18 a + 14\right)\cdot 31^{8} + \left(14 a^{5} + 11 a^{4} + 22 a^{3} + 26 a^{2} + 21 a + 4\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 9 }$ | $=$ | \( 7 a^{5} + 28 a^{4} + 27 a^{3} + 28 a^{2} + 28 a + 29 + \left(27 a^{5} + 22 a^{4} + 26 a^{3} + 22 a^{2} + 23 a + 19\right)\cdot 31 + \left(3 a^{5} + 15 a^{4} + 3 a^{3} + 22 a^{2} + 29 a + 4\right)\cdot 31^{2} + \left(23 a^{5} + 22 a^{4} + 9 a^{2} + 16 a + 7\right)\cdot 31^{3} + \left(5 a^{5} + 10 a^{4} + 30 a^{3} + 6 a^{2} + 20\right)\cdot 31^{4} + \left(11 a^{5} + 4 a^{4} + 25 a^{3} + 9 a^{2} + 29 a + 5\right)\cdot 31^{5} + \left(10 a^{5} + 22 a^{4} + 22 a^{3} + 22 a^{2} + 29 a + 10\right)\cdot 31^{6} + \left(4 a^{5} + 28 a^{4} + 28 a^{3} + 8 a^{2} + 4 a + 29\right)\cdot 31^{7} + \left(13 a^{4} + 15 a^{3} + 15 a^{2} + 10 a + 9\right)\cdot 31^{8} + \left(10 a^{5} + 9 a^{4} + 5 a^{3} + 8 a^{2} + 16 a + 24\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 10 }$ | $=$ | \( 18 a^{4} + 13 a^{3} + 12 a^{2} + a + 5 + \left(28 a^{5} + 2 a^{4} + 28 a^{3} + a^{2} + 21 a + 24\right)\cdot 31 + \left(9 a^{5} + 2 a^{4} + 16 a^{2} + 24 a + 12\right)\cdot 31^{2} + \left(19 a^{5} + 16 a^{4} + 18 a^{3} + 22 a^{2} + 26 a + 12\right)\cdot 31^{3} + \left(11 a^{5} + 17 a^{4} + 23 a^{3} + 28 a^{2} + 19 a + 22\right)\cdot 31^{4} + \left(20 a^{5} + 14 a^{4} + 7 a^{3} + 12 a^{2} + 16 a + 8\right)\cdot 31^{5} + \left(28 a^{5} + 29 a^{4} + 25 a^{3} + 14 a^{2} + 13 a + 17\right)\cdot 31^{6} + \left(18 a^{5} + 8 a^{4} + 20 a^{3} + 9 a^{2} + a + 16\right)\cdot 31^{7} + \left(20 a^{5} + 10 a^{4} + 22 a^{3} + 7 a^{2} + 25 a + 14\right)\cdot 31^{8} + \left(5 a^{5} + 23 a^{4} + 29 a^{3} + a^{2} + 21 a + 15\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 11 }$ | $=$ | \( 16 a^{5} + 25 a^{4} + 22 a^{3} + 13 a^{2} + 15 a + 25 + \left(22 a^{5} + 16 a^{4} + 30 a^{3} + 15 a^{2} + 14 a + 16\right)\cdot 31 + \left(16 a^{5} + 13 a^{4} + 9 a^{3} + 24 a^{2} + 16\right)\cdot 31^{2} + \left(25 a^{5} + 29 a^{4} + 14 a^{3} + 27 a^{2} + 15 a + 30\right)\cdot 31^{3} + \left(5 a^{5} + 21 a^{4} + 28 a^{3} + a^{2} + 3 a + 6\right)\cdot 31^{4} + \left(6 a^{5} + a^{4} + 8 a^{3} + 12 a^{2} + 4 a + 30\right)\cdot 31^{5} + \left(14 a^{5} + 14 a^{4} + 15 a^{3} + 29 a^{2} + 4 a + 6\right)\cdot 31^{6} + \left(14 a^{5} + 27 a^{4} + 12 a^{3} + 18 a^{2} + 9 a + 27\right)\cdot 31^{7} + \left(5 a^{5} + 12 a^{4} + 9 a^{3} + 17 a^{2} + 5 a + 18\right)\cdot 31^{8} + \left(4 a^{5} + 23 a^{4} + 23 a^{3} + 26 a^{2} + 19 a + 29\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 12 }$ | $=$ | \( 12 a^{5} + 17 a^{4} + 16 a^{3} + 28 a^{2} + 18 a + 11 + \left(25 a^{5} + 10 a^{4} + 9 a^{3} + 30 a^{2} + 26 a\right)\cdot 31 + \left(13 a^{5} + 30 a^{4} + 15 a^{3} + 21 a^{2} + 30 a + 15\right)\cdot 31^{2} + \left(15 a^{5} + 10 a^{4} + 29 a^{3} + 2 a^{2} + 14 a + 3\right)\cdot 31^{3} + \left(18 a^{5} + 5 a^{3} + 12 a^{2} + 29 a + 14\right)\cdot 31^{4} + \left(26 a^{5} + a^{4} + 11 a^{3} + 7 a^{2} + 18 a + 25\right)\cdot 31^{5} + \left(14 a^{5} + 24 a^{4} + 18 a^{3} + 29 a^{2} + 2 a + 17\right)\cdot 31^{6} + \left(7 a^{5} + 9 a^{4} + 5 a^{3} + 18 a^{2} + 10 a + 5\right)\cdot 31^{7} + \left(28 a^{5} + 15 a^{4} + 24 a^{3} + 9 a^{2} + 13 a + 1\right)\cdot 31^{8} + \left(26 a^{5} + 12 a^{4} + 7 a^{3} + 28 a^{2} + 7 a + 5\right)\cdot 31^{9} +O(31^{10})\) |
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 |
$1$ | $1$ | $()$ | $11$ |
$55$ | $2$ | $(1,3)(2,9)(4,6)(5,10)(7,8)(11,12)$ | $-1$ |
$66$ | $2$ | $(2,12)(4,5)(6,10)(7,8)(9,11)$ | $-1$ |
$110$ | $3$ | $(1,7,3)(2,6,10)(4,5,11)(8,12,9)$ | $-1$ |
$110$ | $4$ | $(1,12,7,3)(2,4,11,9)(5,8,10,6)$ | $1$ |
$132$ | $5$ | $(2,11,10,7,4)(5,12,9,6,8)$ | $1$ |
$132$ | $5$ | $(1,10,7,4,3)(2,8,11,9,12)$ | $1$ |
$110$ | $6$ | $(1,8,10,7,2,4)(3,12,5,9,6,11)$ | $-1$ |
$132$ | $10$ | $(2,8,11,5,10,12,7,9,4,6)$ | $-1$ |
$132$ | $10$ | $(2,5,7,6,11,12,4,8,10,9)$ | $-1$ |
$120$ | $11$ | $(1,8,11,6,5,12,2,4,10,9,7)$ | $0$ |
$110$ | $12$ | $(1,12,8,5,10,9,7,6,2,11,4,3)$ | $1$ |
$110$ | $12$ | $(1,9,4,5,2,12,7,3,10,11,8,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.