Properties

Label 11.713...000.12t218.a.a
Dimension $11$
Group $\PGL(2,11)$
Conductor $7.133\times 10^{20}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $11$
Group: $\PGL(2,11)$
Conductor: \(713\!\cdots\!000\)\(\medspace = 2^{8} \cdot 5^{10} \cdot 11^{11}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 12.2.713279176527500000000.1
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 12.2.713279176527500000000.1

Defining polynomial

$f(x)$$=$ \( x^{12} - 4 x^{11} + 55 x^{9} - 110 x^{8} - 220 x^{7} + 1155 x^{6} - 440 x^{5} - 4730 x^{4} + 6985 x^{3} + 9460 x^{2} - 29660 x + 18540 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{6} + x^{4} + 9x^{3} + 9x^{2} + x + 5 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.