Properties

Label 11.197...504.24t2949.a.a
Dimension $11$
Group $\PGL(2,11)$
Conductor $1.970\times 10^{28}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $11$
Group: $\PGL(2,11)$
Conductor: \(197\!\cdots\!504\)\(\medspace = 2^{10} \cdot 11^{12} \cdot 19^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 12.2.1791247110799293768894884864.1
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 12.2.1791247110799293768894884864.1

Defining polynomial

$f(x)$$=$ \( x^{12} - x^{11} - 11 x^{10} + 55 x^{9} - 396 x^{8} - 1782 x^{7} + 12518 x^{6} + 64856 x^{5} - 33033 x^{4} - 349701 x^{3} - 1173073 x^{2} - 503293 x - 917874 \) 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:
$r_{ 1 }$ $=$ \( 9 a^{5} + 2 a^{4} + 26 a^{3} + 19 a^{2} + 22 a + 19 + \left(21 a^{5} + 9 a^{4} + 13 a^{3} + 5 a + 26\right)\cdot 31 + \left(27 a^{5} + 7 a^{4} + 14 a^{3} + 10 a^{2} + 2 a + 3\right)\cdot 31^{2} + \left(13 a^{5} + 6 a^{4} + 22 a^{3} + 14 a^{2} + 21 a + 11\right)\cdot 31^{3} + \left(29 a^{5} + 16 a^{4} + 17 a^{3} + 14 a\right)\cdot 31^{4} + \left(11 a^{5} + 12 a^{4} + 10 a^{3} + 22 a^{2} + 30 a + 2\right)\cdot 31^{5} + \left(21 a^{5} + 3 a^{4} + 18 a^{3} + 12 a^{2} + 5\right)\cdot 31^{6} + \left(12 a^{5} + 9 a^{4} + 9 a^{3} + 19 a^{2} + 11 a + 9\right)\cdot 31^{7} + \left(16 a^{5} + 20 a^{4} + 7 a^{3} + 4 a^{2} + 15 a + 20\right)\cdot 31^{8} + \left(25 a^{5} + 25 a^{4} + 18 a^{3} + 13 a^{2} + 13 a + 16\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 a^{5} + 26 a^{4} + 8 a^{3} + 6 a^{2} + 20 a + 27 + \left(28 a^{5} + 18 a^{4} + 12 a^{3} + 21 a^{2} + 10 a + 3\right)\cdot 31 + \left(15 a^{5} + 13 a^{4} + 11 a^{3} + 15 a^{2} + 27 a + 6\right)\cdot 31^{2} + \left(20 a^{5} + 9 a^{4} + 10 a^{3} + 7 a^{2} + 3 a + 10\right)\cdot 31^{3} + \left(10 a^{5} + 2 a^{4} + 13 a^{3} + 15 a^{2} + 18 a + 10\right)\cdot 31^{4} + \left(4 a^{5} + 14 a^{4} + 20 a^{3} + 21 a^{2} + 18\right)\cdot 31^{5} + \left(19 a^{5} + 26 a^{4} + 7 a^{3} + 14 a^{2} + 18 a + 24\right)\cdot 31^{6} + \left(9 a^{5} + 8 a^{4} + 25 a^{3} + 23 a^{2} + 24 a\right)\cdot 31^{7} + \left(13 a^{5} + 7 a^{4} + 20 a^{3} + 20 a^{2} + 4 a + 21\right)\cdot 31^{8} + \left(5 a^{5} + 15 a^{4} + 24 a^{3} + 8 a^{2} + 11 a + 15\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 26 a^{5} + 12 a^{4} + 15 a^{3} + 9 a^{2} + 11 a + 30 + \left(27 a^{5} + 7 a^{4} + 3 a^{3} + 30 a^{2} + 6 a + 20\right)\cdot 31 + \left(18 a^{5} + 4 a^{4} + 9 a^{3} + 20 a^{2} + 7 a + 7\right)\cdot 31^{2} + \left(18 a^{5} + 3 a^{4} + 22 a^{3} + 2 a^{2} + 28 a + 2\right)\cdot 31^{3} + \left(5 a^{5} + 2 a^{4} + 26 a^{3} + 3 a^{2} + 14 a + 24\right)\cdot 31^{4} + \left(30 a^{5} + 19 a^{4} + 2 a^{3} + 12 a^{2} + 26 a + 9\right)\cdot 31^{5} + \left(15 a^{5} + 8 a^{4} + 17 a^{3} + 23 a^{2} + 2 a + 11\right)\cdot 31^{6} + \left(21 a^{5} + 29 a^{4} + 29 a^{3} + 26 a^{2} + 13 a + 13\right)\cdot 31^{7} + \left(27 a^{5} + 23 a^{4} + 7 a^{2} + 28\right)\cdot 31^{8} + \left(20 a^{5} + 17 a^{4} + 6 a^{3} + 8 a^{2} + 24 a + 4\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 a^{5} + 19 a^{4} + 20 a^{3} + 21 a^{2} + 16 a + 17 + \left(20 a^{5} + 21 a^{4} + 30 a^{3} + 23 a^{2} + 6 a + 21\right)\cdot 31 + \left(25 a^{5} + 5 a^{4} + 14 a^{3} + 12 a^{2} + 27 a + 15\right)\cdot 31^{2} + \left(15 a^{5} + 29 a^{4} + 10 a^{3} + 21 a^{2} + 30\right)\cdot 31^{3} + \left(29 a^{5} + 5 a^{3} + 5 a^{2} + 30 a + 29\right)\cdot 31^{4} + \left(15 a^{5} + 22 a^{4} + 20 a^{3} + 15 a^{2} + 11 a + 17\right)\cdot 31^{5} + \left(14 a^{5} + 14 a^{4} + 13 a^{3} + 23 a^{2} + 26 a + 23\right)\cdot 31^{6} + \left(22 a^{5} + 3 a^{4} + 4 a^{3} + 15 a + 27\right)\cdot 31^{7} + \left(29 a^{4} + 7 a^{3} + 17 a^{2} + 5 a + 3\right)\cdot 31^{8} + \left(24 a^{5} + 14 a^{4} + 10 a^{3} + 9 a^{2} + 3 a + 25\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{5} + 10 a^{4} + 19 a^{3} + 21 a^{2} + 3 a + 3 + \left(30 a^{5} + 27 a^{4} + 22 a^{3} + 10 a^{2} + 6 a + 12\right)\cdot 31 + \left(16 a^{5} + 4 a^{4} + 12 a^{3} + 11 a^{2} + 7 a + 14\right)\cdot 31^{2} + \left(20 a^{5} + 12 a^{4} + 21 a^{3} + 17 a^{2} + 7 a + 10\right)\cdot 31^{3} + \left(14 a^{5} + 2 a^{4} + a^{3} + 28 a^{2} + 2 a + 10\right)\cdot 31^{4} + \left(3 a^{5} + 8 a^{4} + 24 a^{3} + 8 a^{2} + 16 a\right)\cdot 31^{5} + \left(25 a^{5} + 17 a^{4} + 11 a^{3} + 17 a^{2} + 16 a + 12\right)\cdot 31^{6} + \left(9 a^{5} + 25 a^{4} + a^{3} + 2 a^{2} + 25\right)\cdot 31^{7} + \left(30 a^{5} + a^{4} + 24 a^{3} + 16 a^{2} + 2 a + 7\right)\cdot 31^{8} + \left(25 a^{5} + 10 a^{4} + 24 a^{2} + 25 a + 23\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 25 a^{4} + 9 a^{3} + 2 a^{2} + 18 a + 17 + \left(25 a^{4} + a^{3} + 10 a^{2} + 19 a + 21\right)\cdot 31 + \left(28 a^{5} + a^{4} + 19 a^{3} + 30 a^{2} + 27 a + 1\right)\cdot 31^{2} + \left(2 a^{5} + 23 a^{4} + 3 a^{3} + 19 a^{2} + 6 a + 26\right)\cdot 31^{3} + \left(7 a^{5} + 9 a^{4} + 14 a^{2} + 3 a + 7\right)\cdot 31^{4} + \left(a^{5} + 16 a^{4} + 20 a^{3} + 13 a^{2} + 9 a + 19\right)\cdot 31^{5} + \left(25 a^{5} + 11 a^{4} + 18 a^{3} + 19 a^{2} + 25 a + 15\right)\cdot 31^{6} + \left(9 a^{5} + 11 a^{4} + 27 a^{3} + 24 a^{2} + 12 a + 9\right)\cdot 31^{7} + \left(16 a^{5} + 7 a^{4} + 18 a^{3} + 13 a^{2} + 11 a + 7\right)\cdot 31^{8} + \left(16 a^{5} + 17 a^{4} + 30 a^{3} + 3 a^{2} + 25 a + 10\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 10 a^{5} + 15 a^{4} + 16 a^{3} + 6 a^{2} + 21 a + 28 + \left(12 a^{5} + 13 a^{4} + a^{3} + 21 a^{2} + 16 a + 20\right)\cdot 31 + \left(12 a^{5} + 14 a^{4} + 30 a^{3} + 9 a^{2} + 3 a + 12\right)\cdot 31^{2} + \left(17 a^{5} + 18 a^{4} + 9 a^{3} + 12 a^{2} + 19 a + 20\right)\cdot 31^{3} + \left(20 a^{5} + 13 a^{3} + 25 a^{2} + 28 a + 6\right)\cdot 31^{4} + \left(30 a^{5} + 19 a^{4} + 5 a^{3} + 18 a^{2} + 26 a + 23\right)\cdot 31^{5} + \left(17 a^{5} + 12 a^{4} + 17 a^{3} + 16 a^{2} + 14 a + 12\right)\cdot 31^{6} + \left(11 a^{5} + 27 a^{4} + 5 a^{3} + 4 a^{2} + 8 a + 29\right)\cdot 31^{7} + \left(8 a^{5} + 11 a^{3} + 21 a^{2} + 30 a + 28\right)\cdot 31^{8} + \left(a^{5} + 8 a^{4} + 10 a^{3} + 26 a^{2} + 28 a + 20\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 2 a^{5} + 7 a^{4} + 18 a^{3} + 11 a^{2} + 7 a + 26 + \left(16 a^{5} + 28 a^{4} + 28 a^{3} + 13 a^{2} + 14 a + 20\right)\cdot 31 + \left(4 a^{5} + 26 a^{4} + 10 a^{3} + 26 a^{2} + 7 a + 10\right)\cdot 31^{2} + \left(3 a^{5} + 11 a^{4} + 7 a^{3} + 16 a^{2} + 24 a + 5\right)\cdot 31^{3} + \left(8 a^{5} + 29 a^{4} + 13 a^{3} + a^{2} + 2 a + 2\right)\cdot 31^{4} + \left(6 a^{5} + 5 a^{4} + 17 a^{3} + 19 a^{2} + 11 a + 14\right)\cdot 31^{5} + \left(16 a^{5} + 26 a^{4} + 30 a^{3} + 8 a^{2} + a + 23\right)\cdot 31^{6} + \left(11 a^{5} + 5 a^{4} + 8 a^{3} + 16 a^{2} + 26 a + 7\right)\cdot 31^{7} + \left(17 a^{5} + 30 a^{4} + a^{3} + 27 a^{2} + a + 24\right)\cdot 31^{8} + \left(15 a^{5} + 22 a^{4} + 19 a^{3} + 3 a^{2} + 4 a + 3\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 12 a^{5} + 24 a^{4} + 5 a^{3} + 9 a^{2} + 11 a + 11 + \left(a^{5} + 11 a^{4} + a^{2} + 24 a + 30\right)\cdot 31 + \left(29 a^{5} + 21 a^{4} + 12 a^{3} + 27 a^{2} + 19 a + 16\right)\cdot 31^{2} + \left(27 a^{5} + 26 a^{4} + 14 a^{3} + 7 a^{2} + a + 23\right)\cdot 31^{3} + \left(28 a^{5} + 21 a^{4} + 9 a^{3} + 11 a^{2} + 11 a + 24\right)\cdot 31^{4} + \left(5 a^{5} + 21 a^{4} + 25 a^{3} + 29 a^{2} + 28 a + 6\right)\cdot 31^{5} + \left(4 a^{5} + 4 a^{4} + 8 a^{3} + 7 a^{2} + 7 a + 24\right)\cdot 31^{6} + \left(8 a^{5} + 27 a^{4} + 22 a^{3} + 3 a^{2} + 11 a + 28\right)\cdot 31^{7} + \left(26 a^{5} + 8 a^{4} + 24 a^{3} + 25 a^{2} + 15 a + 21\right)\cdot 31^{8} + \left(25 a^{5} + 9 a^{4} + 13 a^{3} + 15 a^{2} + 11 a + 5\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 14 a^{5} + 5 a^{4} + 27 a^{3} + 6 a^{2} + 28 a + 10 + \left(29 a^{5} + 5 a^{4} + 17 a^{3} + 3 a^{2} + 30 a + 10\right)\cdot 31 + \left(29 a^{5} + 21 a^{4} + 3 a^{3} + 21 a^{2} + 3 a + 23\right)\cdot 31^{2} + \left(6 a^{5} + 12 a^{4} + 28 a^{3} + 5 a^{2} + 3 a + 19\right)\cdot 31^{3} + \left(10 a^{5} + 5 a^{4} + 24 a^{3} + 25 a^{2} + 16 a\right)\cdot 31^{4} + \left(30 a^{5} + 10 a^{4} + 6 a^{3} + 26 a^{2} + 14 a + 5\right)\cdot 31^{5} + \left(22 a^{5} + 12 a^{4} + 15 a^{3} + 14 a^{2} + 5 a + 28\right)\cdot 31^{6} + \left(19 a^{5} + 18 a^{4} + 2 a^{3} + 22 a^{2} + 2 a + 15\right)\cdot 31^{7} + \left(7 a^{5} + 24 a^{4} + 7 a^{3} + 25 a^{2} + 3 a + 11\right)\cdot 31^{8} + \left(a^{5} + 12 a^{4} + 19 a^{3} + 15 a^{2} + 8 a + 13\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 13 a^{5} + 6 a^{4} + 9 a^{2} + 25 a + 17 + \left(16 a^{5} + a^{4} + 5 a^{3} + 27 a^{2} + 13 a + 22\right)\cdot 31 + \left(7 a^{5} + 25 a^{4} + 29 a^{3} + 8 a^{2} + 25 a + 17\right)\cdot 31^{2} + \left(17 a^{5} + 26 a^{4} + 24 a^{3} + 17 a + 2\right)\cdot 31^{3} + \left(13 a^{5} + 17 a^{4} + 25 a^{3} + 30 a + 25\right)\cdot 31^{4} + \left(29 a^{5} + 18 a^{4} + 3 a^{3} + 26 a^{2} + 7 a + 15\right)\cdot 31^{5} + \left(18 a^{5} + 19 a^{4} + a^{3} + 29 a^{2} + 8 a + 13\right)\cdot 31^{6} + \left(25 a^{5} + 2 a^{4} + 24 a^{3} + 17 a^{2} + 14 a + 8\right)\cdot 31^{7} + \left(21 a^{5} + 25 a^{4} + 30 a^{3} + 5 a^{2} + 20 a + 15\right)\cdot 31^{8} + \left(17 a^{5} + 29 a^{4} + 9 a^{3} + 12 a^{2} + 22 a + 23\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 27 a^{5} + 4 a^{4} + 23 a^{3} + 5 a^{2} + 4 a + 13 + \left(12 a^{5} + 16 a^{4} + 17 a^{3} + 23 a^{2} + 5\right)\cdot 31 + \left(8 a^{4} + 18 a^{3} + 22 a^{2} + 27 a + 24\right)\cdot 31^{2} + \left(21 a^{5} + 6 a^{4} + 10 a^{3} + 28 a^{2} + 20 a + 23\right)\cdot 31^{3} + \left(7 a^{5} + 15 a^{4} + 3 a^{3} + 23 a^{2} + 13 a + 12\right)\cdot 31^{4} + \left(16 a^{5} + 18 a^{4} + 29 a^{3} + 3 a^{2} + 2 a + 22\right)\cdot 31^{5} + \left(15 a^{5} + 28 a^{4} + 25 a^{3} + 28 a^{2} + 27 a + 22\right)\cdot 31^{6} + \left(23 a^{5} + 16 a^{4} + 24 a^{3} + 23 a^{2} + 14 a + 9\right)\cdot 31^{7} + \left(30 a^{5} + 6 a^{4} + 13 a + 26\right)\cdot 31^{8} + \left(5 a^{5} + 2 a^{4} + 23 a^{3} + 13 a^{2} + 8 a + 22\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.