Properties

Label 12.331...609.18t206.a.a
Dimension $12$
Group $S_3 \wr C_3 $
Conductor $3.313\times 10^{17}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3 \wr C_3 $
Conductor: \(331258496613663609\)\(\medspace = 3^{26} \cdot 19^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.5.22082967873.1
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 9.5.22082967873.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 9x^{7} + 27x^{5} - 36x^{3} + 27x - 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.