Properties

Label 11.110...024.24t2949.a.a
Dimension $11$
Group $\PGL(2,11)$
Conductor $1.108\times 10^{26}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $11$
Group: $\PGL(2,11)$
Conductor: \(110\!\cdots\!024\)\(\medspace = 2^{8} \cdot 11^{12} \cdot 13^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 12.2.10069154974041885785533184.1
Galois orbit size: $1$
Smallest permutation container: 24T2949
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\PSL(2,11).C_2$
Projective stem field: Galois closure of 12.2.10069154974041885785533184.1

Defining polynomial

$f(x)$$=$ \( x^{12} - 2 x^{11} - 44 x^{10} + 44 x^{9} + 847 x^{8} - 924 x^{7} - 8844 x^{6} + 19272 x^{5} + 20075 x^{4} - 156442 x^{3} + 83600 x^{2} + 359236 x - 470323 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{6} + 2x^{4} + 18x^{3} + 38x^{2} + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 43 a^{5} + 32 a^{4} + 23 a^{3} + 58 a^{2} + 45 a + 57 + \left(49 a^{5} + 18 a^{4} + 27 a^{3} + 13 a^{2} + 30 a + 24\right)\cdot 59 + \left(22 a^{5} + 33 a^{4} + 13 a^{3} + 53 a^{2} + 51 a + 20\right)\cdot 59^{2} + \left(23 a^{5} + 10 a^{3} + 20 a^{2} + 53 a + 5\right)\cdot 59^{3} + \left(22 a^{5} + 5 a^{4} + 37 a^{3} + 37 a^{2} + 20\right)\cdot 59^{4} + \left(50 a^{5} + 22 a^{4} + 44 a^{3} + a^{2} + 20 a + 40\right)\cdot 59^{5} + \left(9 a^{5} + 35 a^{4} + 36 a^{3} + 33 a^{2} + a + 53\right)\cdot 59^{6} + \left(26 a^{5} + 41 a^{4} + 31 a^{3} + 8 a^{2} + 5 a + 51\right)\cdot 59^{7} + \left(25 a^{5} + 24 a^{4} + 50 a^{2} + 28 a + 31\right)\cdot 59^{8} + \left(15 a^{5} + 27 a^{4} + 41 a^{3} + 48 a^{2} + 17 a + 52\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a^{5} + 28 a^{4} + 43 a^{3} + 26 a^{2} + 8 a + 3 + \left(26 a^{5} + 36 a^{4} + 30 a^{3} + 29 a^{2} + 29 a + 27\right)\cdot 59 + \left(10 a^{5} + 8 a^{4} + 21 a^{3} + 45 a^{2} + 54 a + 24\right)\cdot 59^{2} + \left(29 a^{5} + 35 a^{4} + 56 a^{3} + 19 a^{2} + 7 a + 53\right)\cdot 59^{3} + \left(8 a^{5} + 32 a^{4} + 58 a^{3} + 36 a^{2} + 35\right)\cdot 59^{4} + \left(3 a^{5} + 38 a^{4} + 14 a^{3} + 20 a^{2} + 5 a + 5\right)\cdot 59^{5} + \left(32 a^{5} + 18 a^{4} + 9 a^{3} + 29 a^{2} + 31 a + 54\right)\cdot 59^{6} + \left(6 a^{5} + 5 a^{4} + 39 a^{3} + 25 a^{2} + 57 a + 25\right)\cdot 59^{7} + \left(31 a^{5} + 49 a^{4} + 36 a^{3} + 58 a^{2} + 35 a + 30\right)\cdot 59^{8} + \left(37 a^{5} + 6 a^{4} + 12 a^{3} + 9 a^{2} + 42 a + 10\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 46 a^{5} + 25 a^{4} + 30 a^{3} + 25 a^{2} + 9 a + 39 + \left(43 a^{5} + 49 a^{4} + 36 a^{3} + 21 a^{2} + 51 a + 48\right)\cdot 59 + \left(42 a^{5} + 25 a^{4} + 57 a^{3} + 16 a^{2} + 57 a + 58\right)\cdot 59^{2} + \left(11 a^{5} + 45 a^{4} + 11 a^{3} + 44 a^{2} + 17 a + 32\right)\cdot 59^{3} + \left(26 a^{5} + a^{4} + 44 a^{3} + 3 a^{2} + 42 a + 34\right)\cdot 59^{4} + \left(52 a^{5} + 18 a^{4} + 7 a^{3} + 41 a^{2} + 52 a + 48\right)\cdot 59^{5} + \left(9 a^{5} + 10 a^{4} + 12 a^{3} + 2 a^{2} + 8 a + 48\right)\cdot 59^{6} + \left(12 a^{5} + 13 a^{4} + 32 a^{3} + 42 a^{2} + 32 a + 12\right)\cdot 59^{7} + \left(9 a^{5} + 15 a^{4} + 49 a^{3} + 45 a + 37\right)\cdot 59^{8} + \left(32 a^{5} + 52 a^{4} + 15 a^{3} + 33 a^{2} + 16 a + 34\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a^{5} + 42 a^{4} + 37 a^{3} + 7 a^{2} + 15 a + 19 + \left(19 a^{5} + 47 a^{4} + 51 a^{3} + 52 a^{2} + 26 a + 2\right)\cdot 59 + \left(36 a^{4} + 53 a^{3} + 20 a^{2} + 12 a + 30\right)\cdot 59^{2} + \left(27 a^{5} + a^{4} + 10 a^{3} + 26 a^{2} + 3 a + 35\right)\cdot 59^{3} + \left(54 a^{5} + 44 a^{4} + 43 a^{3} + 36 a + 12\right)\cdot 59^{4} + \left(49 a^{5} + 36 a^{4} + 50 a^{3} + 44 a^{2} + 12 a + 18\right)\cdot 59^{5} + \left(49 a^{5} + 31 a^{4} + 24 a^{3} + 41 a^{2} + 3 a + 33\right)\cdot 59^{6} + \left(22 a^{5} + 57 a^{4} + 31 a^{3} + 49 a^{2} + 46 a + 27\right)\cdot 59^{7} + \left(17 a^{5} + 14 a^{4} + 58 a^{3} + 45 a^{2} + 8 a + 34\right)\cdot 59^{8} + \left(16 a^{5} + 36 a^{4} + 32 a^{3} + 38 a^{2} + 21 a + 48\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 44 a^{5} + 30 a^{4} + 2 a^{3} + 12 a^{2} + 37 a + 15 + \left(54 a^{5} + 43 a^{4} + 23 a^{3} + 45 a^{2} + 40 a + 42\right)\cdot 59 + \left(18 a^{5} + 14 a^{4} + 12 a^{3} + 19 a^{2} + 39 a + 33\right)\cdot 59^{2} + \left(47 a^{5} + 33 a^{4} + 6 a^{3} + 21 a^{2} + 7 a + 55\right)\cdot 59^{3} + \left(30 a^{5} + 50 a^{4} + 57 a^{3} + 13 a^{2} + 48 a + 1\right)\cdot 59^{4} + \left(31 a^{5} + 10 a^{4} + 28 a^{3} + 16 a^{2} + 21 a + 48\right)\cdot 59^{5} + \left(42 a^{5} + 35 a^{3} + 34 a^{2} + a + 7\right)\cdot 59^{6} + \left(36 a^{5} + 19 a^{4} + 16 a^{3} + 50 a^{2} + 51 a + 10\right)\cdot 59^{7} + \left(57 a^{5} + 35 a^{4} + 22 a^{3} + 22 a^{2} + 9 a + 10\right)\cdot 59^{8} + \left(38 a^{5} + 12 a^{4} + 54 a^{3} + 20 a^{2} + 17 a + 16\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 20 a^{5} + 12 a^{4} + 17 a^{3} + 45 a^{2} + 30 a + 10 + \left(34 a^{5} + 19 a^{4} + 5 a^{3} + 14 a^{2} + 56 a + 29\right)\cdot 59 + \left(55 a^{5} + 19 a^{4} + 54 a^{3} + 42 a^{2} + 3 a + 53\right)\cdot 59^{2} + \left(43 a^{5} + 42 a^{4} + 12 a^{3} + 30 a^{2} + 58 a + 13\right)\cdot 59^{3} + \left(21 a^{5} + 5 a^{4} + 21 a^{3} + 20 a^{2} + 49 a + 8\right)\cdot 59^{4} + \left(43 a^{5} + 3 a^{4} + 41 a^{3} + 58 a^{2} + 54 a + 1\right)\cdot 59^{5} + \left(50 a^{5} + 57 a^{4} + 54 a^{3} + 43 a^{2} + 8 a + 2\right)\cdot 59^{6} + \left(20 a^{5} + 25 a^{4} + 37 a^{3} + 32 a^{2} + 24 a + 6\right)\cdot 59^{7} + \left(54 a^{5} + 8 a^{4} + 53 a^{3} + 34 a^{2} + 3 a + 15\right)\cdot 59^{8} + \left(46 a^{5} + 48 a^{4} + 41 a^{3} + 43 a^{2} + 2 a + 49\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 34 a^{5} + 37 a^{4} + 48 a^{3} + 53 a^{2} + 45 a + \left(30 a^{5} + 48 a^{4} + 40 a^{3} + 25 a^{2} + 3 a + 54\right)\cdot 59 + \left(7 a^{5} + 40 a^{4} + 29 a^{3} + 25 a^{2} + 44 a + 20\right)\cdot 59^{2} + \left(31 a^{5} + 7 a^{4} + 46 a^{3} + 20 a^{2} + 27 a + 34\right)\cdot 59^{3} + \left(45 a^{5} + 24 a^{4} + 2 a^{3} + 7 a^{2} + 35 a + 45\right)\cdot 59^{4} + \left(30 a^{5} + 16 a^{4} + 4 a^{3} + 16 a^{2} + 35 a + 18\right)\cdot 59^{5} + \left(13 a^{5} + 2 a^{4} + 12 a^{3} + 9 a^{2} + 18 a + 32\right)\cdot 59^{6} + \left(54 a^{5} + 8 a^{4} + 54 a^{3} + 35 a^{2} + 18 a + 41\right)\cdot 59^{7} + \left(27 a^{5} + 5 a^{4} + 12 a^{3} + 33 a^{2} + 24 a + 40\right)\cdot 59^{8} + \left(33 a^{5} + 25 a^{4} + 8 a^{3} + 10 a^{2} + 28 a + 19\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 5 a^{5} + 17 a^{4} + 36 a^{3} + 52 a^{2} + 49 a + 6 + \left(10 a^{5} + 54 a^{4} + 51 a^{3} + 19 a^{2} + 6 a + 52\right)\cdot 59 + \left(55 a^{5} + 9 a^{4} + 28 a^{3} + 34 a^{2} + 36 a + 50\right)\cdot 59^{2} + \left(51 a^{5} + 12 a^{4} + 7 a^{3} + 38 a + 57\right)\cdot 59^{3} + \left(37 a^{5} + 42 a^{4} + 27 a^{3} + 54 a^{2} + 38 a + 14\right)\cdot 59^{4} + \left(32 a^{5} + 15 a^{4} + 11 a^{3} + 10 a^{2} + 12 a + 9\right)\cdot 59^{5} + \left(15 a^{5} + 17 a^{4} + 36 a^{3} + 14 a^{2} + 33 a + 40\right)\cdot 59^{6} + \left(47 a^{5} + 32 a^{4} + 8 a^{3} + 3 a^{2} + 54 a + 7\right)\cdot 59^{7} + \left(13 a^{5} + 32 a^{4} + 20 a^{3} + 3 a^{2} + 17 a + 26\right)\cdot 59^{8} + \left(49 a^{5} + 56 a^{4} + 41 a^{3} + 36 a^{2} + 29 a + 7\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 24 a^{5} + 51 a^{4} + 52 a^{3} + 8 a^{2} + 54 a + 15 + \left(18 a^{5} + 44 a^{4} + 49 a^{3} + 49 a^{2} + 24 a + 13\right)\cdot 59 + \left(44 a^{5} + 43 a^{4} + 11 a^{3} + 4 a^{2} + 15 a + 22\right)\cdot 59^{2} + \left(10 a^{5} + 42 a^{4} + 33 a^{3} + 11 a^{2} + 8 a + 48\right)\cdot 59^{3} + \left(34 a^{5} + 44 a^{4} + 41 a^{3} + 47 a^{2} + 49 a + 24\right)\cdot 59^{4} + \left(55 a^{5} + 22 a^{4} + 6 a^{3} + 17 a^{2} + 34 a + 23\right)\cdot 59^{5} + \left(30 a^{5} + 12 a^{4} + 53 a^{3} + 17 a^{2} + 25 a + 42\right)\cdot 59^{6} + \left(35 a^{5} + 2 a^{4} + 11 a^{3} + 53 a^{2} + 16 a + 53\right)\cdot 59^{7} + \left(3 a^{5} + 53 a^{4} + 11 a^{3} + 32 a^{2} + 31 a + 41\right)\cdot 59^{8} + \left(25 a^{5} + 44 a^{4} + 45 a^{3} + 21 a + 57\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 27 a^{5} + 14 a^{4} + 23 a^{3} + 11 a^{2} + 3 a + 17 + \left(45 a^{5} + 3 a^{4} + 2 a^{3} + 43 a^{2} + 30 a + 13\right)\cdot 59 + \left(34 a^{5} + 52 a^{4} + 27 a^{3} + 10 a^{2} + 46 a + 58\right)\cdot 59^{2} + \left(9 a^{5} + 47 a^{4} + 10 a^{3} + 56 a^{2} + 26 a + 31\right)\cdot 59^{3} + \left(43 a^{5} + 18 a^{4} + 21 a^{3} + 47 a^{2} + 44 a + 12\right)\cdot 59^{4} + \left(12 a^{5} + 41 a^{4} + 55 a^{3} + a^{2} + 46 a + 18\right)\cdot 59^{5} + \left(a^{5} + 4 a^{4} + a^{3} + 36 a^{2} + 58 a + 26\right)\cdot 59^{6} + \left(28 a^{5} + 16 a^{4} + 2 a^{3} + 30 a^{2} + 33 a + 55\right)\cdot 59^{7} + \left(44 a^{5} + 43 a^{4} + 58 a^{3} + 42 a^{2} + 13 a + 31\right)\cdot 59^{8} + \left(30 a^{5} + 30 a^{4} + 57 a^{3} + 15 a^{2} + 29 a + 6\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 50 a^{5} + 33 a^{4} + 25 a^{3} + 35 a^{2} + 52 a + 31 + \left(5 a^{5} + 34 a^{4} + 3 a^{3} + 42 a^{2} + 6 a + 34\right)\cdot 59 + \left(3 a^{5} + 10 a^{4} + 12 a^{3} + 57 a^{2} + 10 a + 57\right)\cdot 59^{2} + \left(53 a^{5} + 29 a^{4} + 50 a^{3} + 34 a^{2} + 41 a + 13\right)\cdot 59^{3} + \left(31 a^{5} + 22 a^{4} + 52 a^{3} + 28 a^{2} + 53 a + 34\right)\cdot 59^{4} + \left(28 a^{5} + 14 a^{4} + 13 a^{3} + 52 a^{2} + 22 a + 51\right)\cdot 59^{5} + \left(4 a^{5} + 44 a^{4} + 47 a^{3} + 56 a^{2} + 32 a + 6\right)\cdot 59^{6} + \left(6 a^{5} + 11 a^{4} + 26 a^{3} + 53 a^{2} + 49 a + 43\right)\cdot 59^{7} + \left(21 a^{5} + 30 a^{4} + 25 a^{3} + 8 a^{2} + a + 18\right)\cdot 59^{8} + \left(2 a^{5} + 48 a^{4} + 19 a^{3} + 51 a^{2} + 42 a + 50\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 33 a^{4} + 18 a^{3} + 22 a^{2} + 7 a + 26 + \left(16 a^{5} + 12 a^{4} + 31 a^{3} + 55 a^{2} + 47 a + 12\right)\cdot 59 + \left(58 a^{5} + 58 a^{4} + 31 a^{3} + 22 a^{2} + 40 a + 41\right)\cdot 59^{2} + \left(14 a^{5} + 55 a^{4} + 38 a^{3} + 8 a^{2} + 3 a + 29\right)\cdot 59^{3} + \left(56 a^{5} + 2 a^{4} + 5 a^{3} + 57 a^{2} + 14 a + 49\right)\cdot 59^{4} + \left(21 a^{5} + 55 a^{4} + 15 a^{3} + 13 a^{2} + 34 a + 11\right)\cdot 59^{5} + \left(34 a^{5} + a^{4} + 30 a^{3} + 35 a^{2} + 12 a + 6\right)\cdot 59^{6} + \left(57 a^{5} + 3 a^{4} + 2 a^{3} + 27 a^{2} + 24 a + 18\right)\cdot 59^{7} + \left(47 a^{5} + 42 a^{4} + 5 a^{3} + 20 a^{2} + 15 a + 35\right)\cdot 59^{8} + \left(25 a^{5} + 23 a^{4} + 42 a^{3} + 45 a^{2} + 27 a\right)\cdot 59^{9} +O(59^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.