Properties

Label 11.276...576.12t218.a.a
Dimension $11$
Group $\PGL(2,11)$
Conductor $2.760\times 10^{20}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $11$
Group: $\PGL(2,11)$
Conductor: \(276\!\cdots\!576\)\(\medspace = 2^{14} \cdot 3^{10} \cdot 11^{11}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 12.2.276027291040300056576.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,11)$
Parity: odd
Determinant: 1.11.2t1.a.a
Projective image: $\PSL(2,11).C_2$
Projective stem field: Galois closure of 12.2.276027291040300056576.1

Defining polynomial

$f(x)$$=$ \( x^{12} - 2x^{11} - 99x^{8} + 132x^{7} - 132x^{6} + 3267x^{4} - 2178x^{3} + 4356x^{2} - 1656x - 35793 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{6} + x^{4} + 82x^{3} + 80x^{2} + 15x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 44 a^{5} + 80 a^{4} + 39 a^{3} + 49 a^{2} + 66 a + 9 + \left(58 a^{5} + 49 a^{4} + 46 a^{3} + 37 a^{2} + 19 a + 45\right)\cdot 89 + \left(22 a^{5} + 29 a^{4} + 21 a^{3} + 74 a^{2} + 74\right)\cdot 89^{2} + \left(4 a^{5} + a^{4} + 2 a^{3} + 69 a^{2} + 59 a + 84\right)\cdot 89^{3} + \left(44 a^{5} + 87 a^{4} + 15 a^{3} + 66 a^{2} + 44 a + 67\right)\cdot 89^{4} + \left(52 a^{5} + 75 a^{4} + 38 a^{3} + 66 a^{2} + 11 a + 74\right)\cdot 89^{5} + \left(19 a^{5} + 26 a^{4} + 65 a^{3} + 28 a^{2} + 58 a + 85\right)\cdot 89^{6} + \left(52 a^{5} + 38 a^{4} + 3 a^{3} + 29 a^{2} + 54 a + 80\right)\cdot 89^{7} + \left(50 a^{5} + 44 a^{4} + 3 a^{3} + 78 a^{2} + 47 a + 11\right)\cdot 89^{8} + \left(46 a^{5} + 58 a^{4} + 49 a^{3} + 8 a^{2} + 11 a + 73\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 68 a^{5} + 2 a^{4} + 13 a^{3} + 48 a^{2} + 45 a + 25 + \left(52 a^{5} + 52 a^{4} + 52 a^{3} + 88 a^{2} + 24 a + 46\right)\cdot 89 + \left(5 a^{5} + 36 a^{4} + 19 a^{3} + 65 a^{2} + 32 a + 56\right)\cdot 89^{2} + \left(17 a^{5} + 15 a^{4} + 85 a^{3} + 18 a^{2} + 38 a + 59\right)\cdot 89^{3} + \left(20 a^{5} + 18 a^{4} + 71 a^{3} + 46 a^{2} + 43 a + 78\right)\cdot 89^{4} + \left(32 a^{5} + 44 a^{4} + 79 a^{3} + 34 a^{2} + 14 a + 61\right)\cdot 89^{5} + \left(13 a^{5} + 18 a^{4} + 31 a^{3} + 21 a^{2} + 52 a + 7\right)\cdot 89^{6} + \left(74 a^{5} + 54 a^{4} + 41 a^{3} + 5 a^{2} + 46 a + 77\right)\cdot 89^{7} + \left(32 a^{5} + 60 a^{4} + 46 a^{3} + 13 a^{2} + 62 a + 88\right)\cdot 89^{8} + \left(43 a^{5} + 39 a^{4} + 41 a^{3} + 69 a^{2} + 7 a + 78\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 a^{5} + 46 a^{4} + 26 a^{3} + 53 a^{2} + 44 a + 10 + \left(38 a^{5} + 12 a^{4} + 35 a^{3} + 35 a + 34\right)\cdot 89 + \left(71 a^{5} + 59 a^{4} + 26 a^{3} + 36 a^{2} + 65 a + 70\right)\cdot 89^{2} + \left(62 a^{5} + 6 a^{4} + 82 a^{3} + 20 a^{2} + 10 a + 52\right)\cdot 89^{3} + \left(76 a^{5} + 45 a^{4} + 10 a^{3} + 41 a^{2} + 20 a + 72\right)\cdot 89^{4} + \left(7 a^{5} + 15 a^{4} + 74 a^{3} + 48 a^{2} + 14 a + 28\right)\cdot 89^{5} + \left(63 a^{5} + 37 a^{4} + 8 a^{3} + 83 a^{2} + 74 a + 55\right)\cdot 89^{6} + \left(33 a^{5} + 52 a^{4} + 6 a^{3} + 34 a^{2} + 81 a + 77\right)\cdot 89^{7} + \left(78 a^{5} + 31 a^{4} + 86 a^{3} + 7 a^{2} + 27 a + 17\right)\cdot 89^{8} + \left(11 a^{5} + 3 a^{4} + 24 a^{3} + 82 a^{2} + 39 a + 53\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 15 a^{5} + 81 a^{4} + 76 a^{3} + 38 a^{2} + 48 a + 21 + \left(62 a^{5} + 47 a^{4} + 41 a^{3} + 29 a^{2} + 71 a + 71\right)\cdot 89 + \left(29 a^{5} + 39 a^{4} + 69 a^{3} + 68 a^{2} + 14 a + 81\right)\cdot 89^{2} + \left(26 a^{5} + 59 a^{4} + 37 a^{3} + 25 a^{2} + 54 a + 15\right)\cdot 89^{3} + \left(83 a^{5} + 79 a^{4} + 9 a^{3} + 32 a^{2} + 25 a + 66\right)\cdot 89^{4} + \left(77 a^{5} + 52 a^{4} + 31 a^{3} + 43 a^{2} + 72 a + 65\right)\cdot 89^{5} + \left(19 a^{5} + 24 a^{4} + 35 a^{2} + 33 a + 40\right)\cdot 89^{6} + \left(27 a^{5} + 68 a^{4} + 8 a^{3} + 48 a^{2} + 33 a + 22\right)\cdot 89^{7} + \left(6 a^{5} + 43 a^{4} + 17 a^{3} + 67 a^{2} + 67 a + 31\right)\cdot 89^{8} + \left(12 a^{5} + 28 a^{4} + 51 a^{3} + 82 a^{2} + 4 a + 48\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 18 a^{5} + 55 a^{4} + 42 a^{3} + 34 a^{2} + 45 a + 32 + \left(55 a^{5} + 24 a^{4} + 8 a^{3} + 2 a + 72\right)\cdot 89 + \left(41 a^{5} + 42 a^{4} + 10 a^{3} + 34 a^{2} + 86 a + 22\right)\cdot 89^{2} + \left(33 a^{5} + 9 a^{4} + 15 a^{3} + 64 a^{2} + 27 a + 13\right)\cdot 89^{3} + \left(39 a^{5} + 25 a^{4} + 69 a^{3} + 45 a^{2} + 2 a + 78\right)\cdot 89^{4} + \left(55 a^{5} + 26 a^{4} + 48 a^{3} + 58 a^{2} + 26 a + 2\right)\cdot 89^{5} + \left(63 a^{5} + 70 a^{4} + 40 a^{3} + 19 a^{2} + 76\right)\cdot 89^{6} + \left(36 a^{5} + 9 a^{4} + 64 a^{3} + 82 a^{2} + 31 a + 23\right)\cdot 89^{7} + \left(8 a^{5} + 22 a^{4} + 26 a^{3} + 48 a^{2} + 12 a + 74\right)\cdot 89^{8} + \left(33 a^{5} + 70 a^{4} + 84 a^{3} + 53 a^{2} + 83 a + 76\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 51 a^{5} + 62 a^{4} + 28 a^{3} + 32 a^{2} + 5 a + 18 + \left(29 a^{5} + 34 a^{4} + a^{3} + 46 a^{2} + 49 a + 83\right)\cdot 89 + \left(18 a^{5} + 2 a^{4} + 88 a^{3} + 77 a^{2} + 51 a + 44\right)\cdot 89^{2} + \left(34 a^{5} + 47 a^{4} + 87 a^{3} + 76 a^{2} + 77 a + 44\right)\cdot 89^{3} + \left(29 a^{5} + 61 a^{4} + 37 a^{3} + 62 a^{2} + 82 a + 62\right)\cdot 89^{4} + \left(44 a^{5} + 51 a^{4} + 31 a^{3} + 9 a^{2} + 72 a + 55\right)\cdot 89^{5} + \left(79 a^{5} + 51 a^{4} + 18 a^{3} + 16 a^{2} + 4 a + 45\right)\cdot 89^{6} + \left(5 a^{5} + 45 a^{4} + 39 a^{3} + 13 a^{2} + 46 a + 76\right)\cdot 89^{7} + \left(65 a^{5} + 84 a^{4} + 28 a^{3} + 18 a^{2} + 86 a + 70\right)\cdot 89^{8} + \left(41 a^{5} + 70 a^{4} + 53 a^{3} + 79 a^{2} + 45 a + 1\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 5 a^{5} + 20 a^{4} + 77 a^{3} + 2 a^{2} + 65 a + \left(29 a^{5} + 42 a^{4} + 81 a^{3} + 71 a^{2} + 16 a + 46\right)\cdot 89 + \left(54 a^{5} + 19 a^{4} + 85 a^{3} + 18 a^{2} + 59 a + 37\right)\cdot 89^{2} + \left(76 a^{5} + 55 a^{4} + 39 a^{3} + 8 a^{2} + 67 a + 41\right)\cdot 89^{3} + \left(52 a^{5} + 13 a^{4} + 80 a^{3} + 83 a^{2} + 51 a + 5\right)\cdot 89^{4} + \left(49 a^{5} + 52 a^{4} + 21 a^{3} + 8 a^{2} + 20 a + 53\right)\cdot 89^{5} + \left(87 a^{5} + 16 a^{4} + 60 a^{3} + 57 a^{2} + 73 a + 8\right)\cdot 89^{6} + \left(34 a^{5} + 46 a^{4} + 88 a^{3} + 71 a^{2} + 41 a + 48\right)\cdot 89^{7} + \left(77 a^{5} + 8 a^{4} + 70 a^{3} + 43 a^{2} + 38 a + 9\right)\cdot 89^{8} + \left(76 a^{5} + 72 a^{4} + 51 a^{3} + 10 a^{2} + 30 a + 76\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 60 a^{5} + 42 a^{4} + 26 a^{2} + 73 a + 34 + \left(75 a^{5} + 84 a^{4} + 34 a^{3} + 27 a^{2} + 78 a + 4\right)\cdot 89 + \left(49 a^{5} + 10 a^{4} + 68 a^{3} + 53 a^{2} + 2 a + 18\right)\cdot 89^{2} + \left(46 a^{5} + 47 a^{4} + 67 a^{3} + 63 a^{2} + 46 a + 75\right)\cdot 89^{3} + \left(40 a^{5} + 77 a^{4} + 29 a^{3} + 43 a^{2} + 45 a + 61\right)\cdot 89^{4} + \left(26 a^{5} + 59 a^{4} + 61 a^{3} + 13 a^{2} + 31 a + 34\right)\cdot 89^{5} + \left(77 a^{5} + 11 a^{4} + 72 a^{3} + 70 a^{2} + 54 a + 49\right)\cdot 89^{6} + \left(30 a^{5} + 53 a^{4} + 25 a^{3} + 68 a^{2} + 67 a + 40\right)\cdot 89^{7} + \left(48 a^{5} + 74 a^{4} + 17 a^{3} + 18 a^{2} + 79 a + 45\right)\cdot 89^{8} + \left(77 a^{5} + 66 a^{4} + 72 a^{3} + 51 a^{2} + 2 a + 1\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 74 a^{5} + 77 a^{4} + 23 a^{3} + 63 a + 54 + \left(12 a^{5} + 25 a^{4} + 17 a^{3} + 42 a^{2} + 2 a + 66\right)\cdot 89 + \left(63 a^{5} + 22 a^{4} + 45 a^{3} + 18 a^{2} + 18 a + 56\right)\cdot 89^{2} + \left(59 a^{5} + 52 a^{4} + 78 a^{3} + 86 a^{2} + 45 a + 60\right)\cdot 89^{3} + \left(32 a^{5} + 25 a^{4} + 58 a^{3} + 74 a^{2} + 61 a + 19\right)\cdot 89^{4} + \left(9 a^{5} + 19 a^{4} + 80 a^{3} + 5 a^{2} + 85 a + 28\right)\cdot 89^{5} + \left(6 a^{5} + 67 a^{4} + 27 a^{3} + 6 a^{2} + 43 a + 67\right)\cdot 89^{6} + \left(41 a^{5} + 22 a^{4} + 12 a^{3} + 57 a^{2} + 63 a + 47\right)\cdot 89^{7} + \left(68 a^{5} + 47 a^{4} + 43 a^{3} + 16 a^{2} + 10 a + 30\right)\cdot 89^{8} + \left(10 a^{5} + 26 a^{4} + 27 a^{3} + 45 a^{2} + 86 a + 25\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 88 a^{5} + 83 a^{4} + 62 a^{3} + 15 a^{2} + 8 a + 53 + \left(5 a^{5} + 8 a^{4} + 76 a^{3} + 48 a^{2} + 34 a + 25\right)\cdot 89 + \left(73 a^{5} + 39 a^{4} + 55 a^{3} + 84 a^{2} + 31 a + 51\right)\cdot 89^{2} + \left(50 a^{5} + 27 a^{4} + 48 a^{3} + 82 a^{2} + 39 a + 56\right)\cdot 89^{3} + \left(45 a^{5} + 86 a^{4} + 48 a^{3} + 54 a^{2} + 38 a + 71\right)\cdot 89^{4} + \left(39 a^{5} + 74 a^{4} + 25 a^{3} + 29 a^{2} + 63 a + 3\right)\cdot 89^{5} + \left(16 a^{5} + 61 a^{4} + 79 a^{3} + 76 a^{2} + 9 a + 72\right)\cdot 89^{6} + \left(46 a^{5} + 25 a^{4} + 69 a^{3} + 65 a^{2} + 69 a + 80\right)\cdot 89^{7} + \left(71 a^{5} + 3 a^{4} + 13 a^{3} + 80 a^{2} + 42 a + 61\right)\cdot 89^{8} + \left(84 a^{5} + 60 a^{4} + 27 a^{3} + 51 a^{2} + 42 a + 30\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 16 a^{5} + 28 a^{4} + 60 a^{3} + 40 a^{2} + 31 a + 46 + \left(88 a^{5} + 12 a^{4} + 17 a^{3} + 43 a^{2} + 31 a + 32\right)\cdot 89 + \left(12 a^{5} + 84 a^{4} + 79 a^{3} + 50 a^{2} + 53 a + 33\right)\cdot 89^{2} + \left(70 a^{5} + 12 a^{4} + 62 a^{3} + 24 a^{2} + 29 a + 50\right)\cdot 89^{3} + \left(43 a^{5} + 49 a^{4} + 8 a^{3} + 3 a^{2} + 73 a + 51\right)\cdot 89^{4} + \left(29 a^{5} + 27 a^{4} + 42 a^{3} + 12 a^{2} + 18 a + 87\right)\cdot 89^{5} + \left(18 a^{5} + 80 a^{4} + 16 a^{3} + 46 a^{2} + 61 a + 50\right)\cdot 89^{6} + \left(15 a^{5} + 36 a^{4} + 53 a^{3} + 25 a^{2} + 24 a + 59\right)\cdot 89^{7} + \left(82 a^{5} + 52 a^{4} + 60 a^{3} + 64 a^{2} + 69 a + 68\right)\cdot 89^{8} + \left(86 a^{5} + 63 a^{4} + 47 a^{3} + 24 a^{2} + 80 a + 72\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 79 a^{5} + 47 a^{4} + 88 a^{3} + 19 a^{2} + 41 a + 56 + \left(25 a^{5} + 49 a^{4} + 31 a^{3} + 10 a^{2} + 78 a + 6\right)\cdot 89 + \left(2 a^{5} + 59 a^{4} + 53 a^{3} + 41 a^{2} + 29 a + 75\right)\cdot 89^{2} + \left(52 a^{5} + 21 a^{4} + 14 a^{3} + 81 a^{2} + 38 a + 67\right)\cdot 89^{3} + \left(25 a^{5} + 54 a^{4} + 4 a^{3} + 67 a^{2} + 44 a + 75\right)\cdot 89^{4} + \left(20 a^{5} + 33 a^{4} + 88 a^{3} + 24 a^{2} + 13 a + 36\right)\cdot 89^{5} + \left(69 a^{5} + 67 a^{4} + 22 a^{3} + 73 a^{2} + 68 a + 63\right)\cdot 89^{6} + \left(46 a^{5} + 80 a^{4} + 32 a^{3} + 31 a^{2} + 62 a + 76\right)\cdot 89^{7} + \left(33 a^{5} + 60 a^{4} + 31 a^{3} + 76 a^{2} + 77 a + 22\right)\cdot 89^{8} + \left(8 a^{5} + 62 a^{4} + 3 a^{3} + 63 a^{2} + 9 a + 84\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

Cycle notation
$(2,8,6,12,3,11,7,4,9,5)$
$(1,10)(2,4)(3,12)(5,9)(6,11)(7,8)$
$(1,8,6,5,12,11,2,9,3,4,7)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character value
$1$$1$$()$$11$
$55$$2$$(1,10)(2,4)(3,12)(5,9)(6,11)(7,8)$$-1$
$66$$2$$(2,11)(3,5)(4,6)(7,8)(9,12)$$1$
$110$$3$$(1,7,10)(2,5,3)(4,8,11)(6,9,12)$$-1$
$110$$4$$(1,11,7,10)(2,9,6,4)(3,5,12,8)$$-1$
$132$$5$$(2,6,3,7,9)(4,5,8,12,11)$$1$
$132$$5$$(1,3,7,9,10)(2,8,6,4,11)$$1$
$110$$6$$(1,8,3,7,2,9)(4,5,6,10,11,12)$$-1$
$132$$10$$(2,8,6,12,3,11,7,4,9,5)$$1$
$132$$10$$(2,12,7,5,6,11,9,8,3,4)$$1$
$120$$11$$(1,8,6,5,12,11,2,9,3,4,7)$$0$
$110$$12$$(1,11,8,12,3,4,7,5,2,6,9,10)$$-1$
$110$$12$$(1,4,9,12,2,11,7,10,3,6,8,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.