Properties

Label 12.223...504.18t206.a.a
Dimension $12$
Group $S_3 \wr C_3 $
Conductor $2.236\times 10^{17}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3 \wr C_3 $
Conductor: \(223580268118933504\)\(\medspace = 2^{18} \cdot 31^{8}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.28400117792.1
Galois orbit size: $1$
Smallest permutation container: 18T206
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_3^3:C_2^2.C_6$
Projective stem field: Galois closure of 9.3.28400117792.1

Defining polynomial

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.