Properties

Label 11.750...576.24t2949.a.a
Dimension $11$
Group $\PGL(2,11)$
Conductor $7.503\times 10^{21}$
Root number $1$
Indicator $1$

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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: $\PSL(2,11).C_2$
Projective stem field: Galois closure of 12.2.82527728843210964110336.1

Defining polynomial

$f(x)$$=$ \( x^{12} - x^{11} - 11x^{10} - 55x^{9} - 66x^{7} - 154x^{6} + 66x^{5} + 165x^{4} + 275x^{3} - 11x^{2} + 21x - 758 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{6} + 19x^{3} + 16x^{2} + 8x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a^{5} + 19 a^{4} + 23 a^{3} + 8 a^{2} + 12 a + 15 + \left(13 a^{5} + 21 a^{3} + a^{2} + 22 a + 2\right)\cdot 31 + \left(27 a^{5} + 10 a^{4} + 2 a^{3} + 14 a^{2} + 22 a + 9\right)\cdot 31^{2} + \left(26 a^{5} + 28 a^{4} + 12 a^{3} + 24 a^{2} + 22 a + 18\right)\cdot 31^{3} + \left(24 a^{5} + 2 a^{4} + 29 a^{3} + 29 a^{2} + 28 a + 25\right)\cdot 31^{4} + \left(18 a^{5} + 28 a^{4} + 19 a^{3} + 27 a^{2} + 19 a + 25\right)\cdot 31^{5} + \left(16 a^{5} + 6 a^{4} + 9 a^{3} + 4 a^{2} + 21 a + 7\right)\cdot 31^{6} + \left(30 a^{5} + 23 a^{4} + 6 a^{3} + 23 a^{2} + 13 a + 10\right)\cdot 31^{7} + \left(11 a^{5} + 6 a^{4} + 21 a^{3} + 15 a^{2} + 16 a + 29\right)\cdot 31^{8} + \left(19 a^{5} + 28 a^{4} + 6 a^{3} + 5 a^{2} + 30 a + 27\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{5} + 5 a^{4} + 3 a^{3} + 9 a^{2} + 5 a + 7 + \left(30 a^{5} + 18 a^{4} + 27 a^{3} + 3 a^{2} + 12 a + 12\right)\cdot 31 + \left(26 a^{5} + 22 a^{4} + 13 a^{3} + 2 a^{2} + 23 a + 3\right)\cdot 31^{2} + \left(a^{5} + 11 a^{4} + 15 a^{3} + 5 a^{2} + 27 a + 6\right)\cdot 31^{3} + \left(25 a^{5} + 17 a^{4} + 12 a^{3} + 13 a^{2} + 10 a + 5\right)\cdot 31^{4} + \left(a^{5} + 16 a^{4} + 16 a^{3} + 11 a^{2} + 28 a + 14\right)\cdot 31^{5} + \left(13 a^{5} + 17 a^{4} + 23 a^{3} + 21 a + 12\right)\cdot 31^{6} + \left(26 a^{5} + 20 a^{4} + 27 a^{2} + 3 a + 15\right)\cdot 31^{7} + \left(10 a^{5} + 19 a^{4} + 9 a^{3} + 20 a^{2} + 16 a + 2\right)\cdot 31^{8} + \left(13 a^{5} + 20 a^{4} + 17 a^{3} + 6 a^{2} + 5 a + 23\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a^{5} + 5 a^{4} + 30 a^{3} + 20 a^{2} + 30 a + 16 + \left(2 a^{5} + 29 a^{4} + 12 a^{3} + 8 a + 21\right)\cdot 31 + \left(28 a^{5} + 16 a^{4} + 18 a^{3} + 21 a^{2} + 13 a + 27\right)\cdot 31^{2} + \left(18 a^{5} + 27 a^{4} + 13 a^{3} + 2 a^{2} + 25 a + 10\right)\cdot 31^{3} + \left(13 a^{5} + 2 a^{4} + 7 a^{3} + 29 a + 12\right)\cdot 31^{4} + \left(25 a^{5} + 2 a^{4} + 28 a^{3} + 10 a^{2} + 3 a + 10\right)\cdot 31^{5} + \left(18 a^{5} + 10 a^{4} + 6 a^{3} + 22 a^{2} + 12 a + 29\right)\cdot 31^{6} + \left(23 a^{5} + 7 a^{4} + 19 a^{3} + 25 a^{2} + 29 a + 14\right)\cdot 31^{7} + \left(9 a^{5} + 19 a^{4} + 13 a^{3} + 24 a^{2} + 23 a + 5\right)\cdot 31^{8} + \left(12 a^{5} + 30 a^{4} + 12 a^{3} + 9 a^{2} + 16\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a^{5} + 28 a^{4} + 11 a^{3} + 10 a^{2} + 10 a + 27 + \left(14 a^{5} + 28 a^{4} + 6 a^{3} + 8 a^{2} + 22 a + 22\right)\cdot 31 + \left(26 a^{5} + 11 a^{4} + 5 a^{3} + 29 a^{2} + 16 a + 15\right)\cdot 31^{2} + \left(19 a^{5} + 26 a^{4} + 26 a^{3} + 22 a^{2} + 26 a + 22\right)\cdot 31^{3} + \left(22 a^{5} + 10 a^{4} + a^{3} + 12 a^{2} + 12 a + 19\right)\cdot 31^{4} + \left(18 a^{5} + 3 a^{4} + 27 a^{3} + 10 a^{2} + 9 a + 4\right)\cdot 31^{5} + \left(24 a^{5} + 18 a^{4} + 29 a^{3} + 9 a^{2} + 10 a + 11\right)\cdot 31^{6} + \left(19 a^{4} + 30 a^{3} + 26 a^{2} + 15 a + 7\right)\cdot 31^{7} + \left(10 a^{5} + 14 a^{4} + 17 a^{3} + 11 a^{2} + 28 a + 18\right)\cdot 31^{8} + \left(6 a^{5} + 21 a^{4} + 26 a^{3} + 6 a^{2} + 28 a + 16\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a^{5} + 11 a^{4} + 29 a^{3} + 5 a^{2} + 7 a + 8 + \left(20 a^{4} + 30 a^{3} + 16 a^{2} + 28 a + 30\right)\cdot 31 + \left(9 a^{5} + 30 a^{4} + 15 a^{3} + 17 a^{2} + 23\right)\cdot 31^{2} + \left(23 a^{4} + 25 a^{3} + 3 a^{2} + 8 a + 13\right)\cdot 31^{3} + \left(29 a^{5} + 3 a^{4} + 3 a^{3} + 2 a^{2} + 20 a + 7\right)\cdot 31^{4} + \left(3 a^{5} + 13 a^{4} + 3 a^{3} + 8 a^{2} + 24 a + 30\right)\cdot 31^{5} + \left(8 a^{5} + 16 a^{4} + 16 a^{3} + 30 a^{2} + 14\right)\cdot 31^{6} + \left(30 a^{5} + 2 a^{4} + 29 a^{3} + 12 a^{2} + 28\right)\cdot 31^{7} + \left(5 a^{5} + 28 a^{4} + 25 a^{3} + 16 a^{2} + 9 a + 10\right)\cdot 31^{8} + \left(20 a^{5} + 4 a^{4} + 15 a^{3} + 29 a^{2} + 16 a + 2\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a^{5} + 7 a^{4} + 30 a^{2} + 26 a + 8 + \left(14 a^{5} + 24 a^{4} + 13 a^{3} + 6 a^{2} + 21 a + 5\right)\cdot 31 + \left(21 a^{5} + 16 a^{4} + 29 a^{3} + 16 a^{2} + 17 a + 14\right)\cdot 31^{2} + \left(9 a^{5} + 9 a^{4} + 9 a^{3} + 15 a^{2} + 12 a + 22\right)\cdot 31^{3} + \left(20 a^{5} + 22 a^{4} + 17 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 31^{4} + \left(19 a^{5} + 8 a^{4} + 15 a^{3} + 15 a^{2} + 12 a + 20\right)\cdot 31^{5} + \left(26 a^{5} + 6 a^{4} + 13 a^{3} + 21 a^{2} + 12 a + 10\right)\cdot 31^{6} + \left(12 a^{5} + 17 a^{4} + 28 a^{3} + 7 a^{2} + 8 a + 17\right)\cdot 31^{7} + \left(12 a^{5} + 3 a^{4} + 3 a^{3} + 9 a^{2} + 6\right)\cdot 31^{8} + \left(19 a^{5} + 12 a^{4} + 28 a^{3} + 21 a^{2} + 2 a + 20\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 13 a^{5} + 15 a^{4} + 19 a^{3} + 15 a^{2} + 15 a + 3 + \left(26 a^{5} + 7 a^{4} + 17 a^{3} + 8 a^{2} + 13 a + 20\right)\cdot 31 + \left(8 a^{5} + 4 a^{4} + 2 a^{3} + 26 a^{2} + 4 a + 18\right)\cdot 31^{2} + \left(30 a^{5} + 16 a^{4} + 5 a^{3} + 29 a^{2} + 18 a + 4\right)\cdot 31^{3} + \left(26 a^{5} + 29 a^{4} + 20 a^{3} + a^{2} + 26 a + 29\right)\cdot 31^{4} + \left(7 a^{5} + 16 a^{4} + 12 a^{3} + 18 a^{2} + 3 a + 17\right)\cdot 31^{5} + \left(15 a^{5} + 26 a^{4} + 14 a^{3} + 4 a + 5\right)\cdot 31^{6} + \left(11 a^{5} + 4 a^{4} + 23 a^{3} + 20 a^{2} + 20 a\right)\cdot 31^{7} + \left(30 a^{5} + 12 a^{4} + 28 a^{3} + 3 a^{2} + 19 a + 23\right)\cdot 31^{8} + \left(2 a^{5} + 18 a^{4} + 20 a^{3} + 16 a^{2} + 15 a\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 27 a^{5} + 8 a^{4} + 24 a^{3} + 8 a^{2} + 19 a + 2 + \left(11 a^{5} + 4 a^{4} + 22 a^{3} + 8 a^{2} + a + 10\right)\cdot 31 + \left(24 a^{5} + 11 a^{4} + 5 a^{3} + 5 a^{2} + a + 24\right)\cdot 31^{2} + \left(25 a^{5} + 25 a^{4} + 16 a^{3} + 19 a^{2} + 2 a + 2\right)\cdot 31^{3} + \left(12 a^{5} + 14 a^{4} + 5 a^{3} + 6 a^{2} + 25 a + 16\right)\cdot 31^{4} + \left(25 a^{5} + 13 a^{4} + 9 a^{3} + 12 a^{2} + 14 a + 23\right)\cdot 31^{5} + \left(25 a^{5} + 25 a^{4} + 21 a^{3} + a^{2} + 21 a + 10\right)\cdot 31^{6} + \left(4 a^{5} + 15 a^{4} + 10 a^{3} + 18 a^{2} + 7 a + 13\right)\cdot 31^{7} + \left(9 a^{5} + 29 a^{4} + 24 a^{3} + 2 a^{2} + 18 a + 14\right)\cdot 31^{8} + \left(14 a^{5} + 11 a^{4} + 22 a^{3} + 26 a^{2} + 21 a + 4\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 7 a^{5} + 28 a^{4} + 27 a^{3} + 28 a^{2} + 28 a + 29 + \left(27 a^{5} + 22 a^{4} + 26 a^{3} + 22 a^{2} + 23 a + 19\right)\cdot 31 + \left(3 a^{5} + 15 a^{4} + 3 a^{3} + 22 a^{2} + 29 a + 4\right)\cdot 31^{2} + \left(23 a^{5} + 22 a^{4} + 9 a^{2} + 16 a + 7\right)\cdot 31^{3} + \left(5 a^{5} + 10 a^{4} + 30 a^{3} + 6 a^{2} + 20\right)\cdot 31^{4} + \left(11 a^{5} + 4 a^{4} + 25 a^{3} + 9 a^{2} + 29 a + 5\right)\cdot 31^{5} + \left(10 a^{5} + 22 a^{4} + 22 a^{3} + 22 a^{2} + 29 a + 10\right)\cdot 31^{6} + \left(4 a^{5} + 28 a^{4} + 28 a^{3} + 8 a^{2} + 4 a + 29\right)\cdot 31^{7} + \left(13 a^{4} + 15 a^{3} + 15 a^{2} + 10 a + 9\right)\cdot 31^{8} + \left(10 a^{5} + 9 a^{4} + 5 a^{3} + 8 a^{2} + 16 a + 24\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 18 a^{4} + 13 a^{3} + 12 a^{2} + a + 5 + \left(28 a^{5} + 2 a^{4} + 28 a^{3} + a^{2} + 21 a + 24\right)\cdot 31 + \left(9 a^{5} + 2 a^{4} + 16 a^{2} + 24 a + 12\right)\cdot 31^{2} + \left(19 a^{5} + 16 a^{4} + 18 a^{3} + 22 a^{2} + 26 a + 12\right)\cdot 31^{3} + \left(11 a^{5} + 17 a^{4} + 23 a^{3} + 28 a^{2} + 19 a + 22\right)\cdot 31^{4} + \left(20 a^{5} + 14 a^{4} + 7 a^{3} + 12 a^{2} + 16 a + 8\right)\cdot 31^{5} + \left(28 a^{5} + 29 a^{4} + 25 a^{3} + 14 a^{2} + 13 a + 17\right)\cdot 31^{6} + \left(18 a^{5} + 8 a^{4} + 20 a^{3} + 9 a^{2} + a + 16\right)\cdot 31^{7} + \left(20 a^{5} + 10 a^{4} + 22 a^{3} + 7 a^{2} + 25 a + 14\right)\cdot 31^{8} + \left(5 a^{5} + 23 a^{4} + 29 a^{3} + a^{2} + 21 a + 15\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 16 a^{5} + 25 a^{4} + 22 a^{3} + 13 a^{2} + 15 a + 25 + \left(22 a^{5} + 16 a^{4} + 30 a^{3} + 15 a^{2} + 14 a + 16\right)\cdot 31 + \left(16 a^{5} + 13 a^{4} + 9 a^{3} + 24 a^{2} + 16\right)\cdot 31^{2} + \left(25 a^{5} + 29 a^{4} + 14 a^{3} + 27 a^{2} + 15 a + 30\right)\cdot 31^{3} + \left(5 a^{5} + 21 a^{4} + 28 a^{3} + a^{2} + 3 a + 6\right)\cdot 31^{4} + \left(6 a^{5} + a^{4} + 8 a^{3} + 12 a^{2} + 4 a + 30\right)\cdot 31^{5} + \left(14 a^{5} + 14 a^{4} + 15 a^{3} + 29 a^{2} + 4 a + 6\right)\cdot 31^{6} + \left(14 a^{5} + 27 a^{4} + 12 a^{3} + 18 a^{2} + 9 a + 27\right)\cdot 31^{7} + \left(5 a^{5} + 12 a^{4} + 9 a^{3} + 17 a^{2} + 5 a + 18\right)\cdot 31^{8} + \left(4 a^{5} + 23 a^{4} + 23 a^{3} + 26 a^{2} + 19 a + 29\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 12 a^{5} + 17 a^{4} + 16 a^{3} + 28 a^{2} + 18 a + 11 + \left(25 a^{5} + 10 a^{4} + 9 a^{3} + 30 a^{2} + 26 a\right)\cdot 31 + \left(13 a^{5} + 30 a^{4} + 15 a^{3} + 21 a^{2} + 30 a + 15\right)\cdot 31^{2} + \left(15 a^{5} + 10 a^{4} + 29 a^{3} + 2 a^{2} + 14 a + 3\right)\cdot 31^{3} + \left(18 a^{5} + 5 a^{3} + 12 a^{2} + 29 a + 14\right)\cdot 31^{4} + \left(26 a^{5} + a^{4} + 11 a^{3} + 7 a^{2} + 18 a + 25\right)\cdot 31^{5} + \left(14 a^{5} + 24 a^{4} + 18 a^{3} + 29 a^{2} + 2 a + 17\right)\cdot 31^{6} + \left(7 a^{5} + 9 a^{4} + 5 a^{3} + 18 a^{2} + 10 a + 5\right)\cdot 31^{7} + \left(28 a^{5} + 15 a^{4} + 24 a^{3} + 9 a^{2} + 13 a + 1\right)\cdot 31^{8} + \left(26 a^{5} + 12 a^{4} + 7 a^{3} + 28 a^{2} + 7 a + 5\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction 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.