Properties

Label 6.19647383.9t28.a.a
Dimension $6$
Group $S_3 \wr C_3 $
Conductor $19647383$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_3 \wr C_3 $
Conductor: \(19647383\)\(\medspace = 7^{6} \cdot 167 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.962721767.1
Galois orbit size: $1$
Smallest permutation container: $S_3 \wr C_3 $
Parity: odd
Determinant: 1.167.2t1.a.a
Projective image: $S_3\wr C_3$
Projective stem field: Galois closure of 9.3.962721767.1

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} + 2x^{7} - 4x^{2} + 4x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 9 a^{2} + 7 a + 6 + \left(6 a^{2} + 2 a + 1\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(10 a^{2} + 9 a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 2 a + 4\right)\cdot 11^{4} + \left(a^{2} + 9\right)\cdot 11^{5} + \left(6 a + 4\right)\cdot 11^{6} + \left(2 a^{2} + 10 a + 8\right)\cdot 11^{7} + \left(8 a^{2} + 5 a + 5\right)\cdot 11^{8} + \left(9 a^{2} + 6 a + 10\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a^{2} + 5 a + 9 + \left(2 a^{2} + 3 a + 7\right)\cdot 11 + \left(4 a + 2\right)\cdot 11^{2} + \left(9 a^{2} + 7 a + 10\right)\cdot 11^{3} + \left(8 a^{2} + 9 a + 5\right)\cdot 11^{4} + \left(3 a^{2} + 9 a + 9\right)\cdot 11^{5} + \left(3 a^{2} + 5 a + 9\right)\cdot 11^{6} + \left(3 a^{2} + 2 a + 6\right)\cdot 11^{7} + \left(5 a^{2} + 10 a + 8\right)\cdot 11^{8} + \left(10 a + 9\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 5 + \left(a^{2} + 10 a + 8\right)\cdot 11 + a\cdot 11^{2} + \left(2 a^{2} + 10\right)\cdot 11^{3} + \left(4 a^{2} + a + 2\right)\cdot 11^{4} + \left(8 a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{6} + \left(3 a^{2} + 10 a + 6\right)\cdot 11^{7} + \left(9 a^{2} + 3\right)\cdot 11^{8} + \left(9 a^{2} + 2 a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a^{2} + 2 a + 8 + \left(2 a^{2} + 8 a\right)\cdot 11 + \left(2 a^{2} + 10 a + 9\right)\cdot 11^{2} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(6 a^{2} + 10\right)\cdot 11^{4} + \left(7 a^{2} + 8 a + 10\right)\cdot 11^{5} + \left(2 a^{2} + 10 a + 8\right)\cdot 11^{6} + \left(9 a^{2} + 8 a + 3\right)\cdot 11^{7} + \left(5 a^{2} + 4 a + 9\right)\cdot 11^{8} + \left(2 a^{2} + 7 a + 8\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a^{2} + 2 a + 4 + \left(3 a^{2} + 9 a\right)\cdot 11 + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{2} + \left(9 a^{2} + 9\right)\cdot 11^{3} + \left(9 a^{2} + 7 a + 6\right)\cdot 11^{4} + \left(4 a + 4\right)\cdot 11^{5} + \left(5 a^{2} + 6 a\right)\cdot 11^{6} + \left(5 a^{2} + 2\right)\cdot 11^{7} + \left(4 a^{2} + 4 a + 8\right)\cdot 11^{8} + \left(2 a^{2} + 2 a\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a^{2} + 3 a + 5 + 6\cdot 11 + \left(6 a^{2} + 10 a + 8\right)\cdot 11^{2} + \left(a^{2} + 8 a + 3\right)\cdot 11^{3} + \left(a^{2} + a + 6\right)\cdot 11^{4} + \left(4 a^{2} + 2 a + 8\right)\cdot 11^{5} + \left(5 a^{2} + 3 a\right)\cdot 11^{6} + \left(9 a + 7\right)\cdot 11^{7} + \left(6 a^{2} + 8 a\right)\cdot 11^{8} + \left(10 a^{2} + 9 a\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 10 a^{2} + 1 + \left(3 a^{2} + 7 a + 4\right)\cdot 11 + \left(2 a^{2} + 8 a + 7\right)\cdot 11^{2} + \left(3 a^{2} + 8 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 8 a + 5\right)\cdot 11^{4} + \left(2 a^{2} + a + 6\right)\cdot 11^{5} + \left(6 a^{2} + 9 a + 5\right)\cdot 11^{6} + \left(3 a^{2} + 7\right)\cdot 11^{7} + \left(7 a^{2} + 10 a + 9\right)\cdot 11^{8} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 7 a^{2} + 4 a + 6 + \left(6 a^{2} + 10 a + 6\right)\cdot 11 + \left(8 a^{2} + 6 a + 6\right)\cdot 11^{2} + \left(3 a^{2} + 5 a + 10\right)\cdot 11^{3} + \left(6 a^{2} + 9\right)\cdot 11^{4} + \left(10 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(4 a^{2} + 5 a + 8\right)\cdot 11^{6} + \left(9 a^{2} + 10 a + 7\right)\cdot 11^{7} + \left(10 a^{2} + 6 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 3 a + 8\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 10 a^{2} + 8 a + 1 + \left(6 a^{2} + 3 a + 8\right)\cdot 11 + \left(2 a^{2} + 3 a + 7\right)\cdot 11^{2} + \left(6 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(3 a^{2} + 2\right)\cdot 11^{4} + \left(4 a^{2} + 7 a + 5\right)\cdot 11^{5} + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{6} + \left(6 a^{2} + 4\right)\cdot 11^{7} + \left(8 a^{2} + 3 a\right)\cdot 11^{8} + \left(a^{2} + 9 a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$6$
$9$$2$$(1,2)$$4$
$27$$2$$(1,2)(3,4)(5,8)$$0$
$27$$2$$(1,2)(8,9)$$2$
$6$$3$$(3,4,6)$$3$
$8$$3$$(1,2,7)(3,4,6)(5,8,9)$$-3$
$12$$3$$(3,4,6)(5,8,9)$$0$
$36$$3$$(1,5,3)(2,8,4)(6,7,9)$$0$
$36$$3$$(1,3,5)(2,4,8)(6,9,7)$$0$
$18$$6$$(1,2)(3,4,6)$$1$
$18$$6$$(1,2)(5,8,9)$$1$
$36$$6$$(1,2)(3,4,6)(5,8,9)$$-2$
$54$$6$$(1,2)(3,4,6)(8,9)$$-1$
$108$$6$$(1,8,4,2,5,3)(6,7,9)$$0$
$108$$6$$(1,3,5,2,4,8)(6,9,7)$$0$
$72$$9$$(1,5,3,2,8,4,7,9,6)$$0$
$72$$9$$(1,3,8,7,6,5,2,4,9)$$0$

The blue line marks the conjugacy class containing complex conjugation.