Properties

Label 12.289...024.36t2210.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $2.892\times 10^{23}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(289\!\cdots\!024\)\(\medspace = 2^{12} \cdot 19^{6} \cdot 107^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.187881728.1
Galois orbit size: $1$
Smallest permutation container: 36T2210
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.1.187881728.1

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} - 2x^{7} + 2x^{5} + 2x^{3} + x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{3} + 7x + 59 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 49 a^{2} + 31 a + 9 + \left(49 a^{2} + 22 a + 5\right)\cdot 61 + \left(41 a^{2} + 29 a + 5\right)\cdot 61^{2} + \left(46 a^{2} + 22 a + 31\right)\cdot 61^{3} + \left(17 a^{2} + 20 a + 28\right)\cdot 61^{4} + \left(36 a^{2} + 33 a + 58\right)\cdot 61^{5} + \left(22 a^{2} + 4 a + 28\right)\cdot 61^{6} + \left(25 a^{2} + 11 a + 37\right)\cdot 61^{7} + \left(57 a^{2} + 29 a + 58\right)\cdot 61^{8} + \left(60 a + 5\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a^{2} + 10 a + 28 + \left(20 a^{2} + 18 a + 21\right)\cdot 61 + \left(46 a^{2} + 2 a + 41\right)\cdot 61^{2} + \left(13 a^{2} + 49 a + 26\right)\cdot 61^{3} + \left(21 a^{2} + 36 a + 60\right)\cdot 61^{4} + \left(52 a^{2} + 49 a + 31\right)\cdot 61^{5} + \left(44 a^{2} + 8 a + 42\right)\cdot 61^{6} + \left(25 a^{2} + 6 a + 29\right)\cdot 61^{7} + \left(44 a^{2} + 24 a + 28\right)\cdot 61^{8} + \left(58 a^{2} + 6 a + 19\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a^{2} + 31 a + 57 + \left(24 a^{2} + 34 a + 5\right)\cdot 61 + \left(43 a^{2} + 7 a + 59\right)\cdot 61^{2} + \left(7 a^{2} + 29 a + 56\right)\cdot 61^{3} + \left(34 a^{2} + 3 a + 48\right)\cdot 61^{4} + \left(29 a^{2} + 27 a + 13\right)\cdot 61^{5} + \left(16 a^{2} + 48 a + 15\right)\cdot 61^{6} + \left(29 a^{2} + 51 a + 23\right)\cdot 61^{7} + \left(55 a^{2} + 27 a + 57\right)\cdot 61^{8} + \left(a^{2} + 19 a + 17\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{2} + 26 a + 21 + \left(33 a^{2} + 52 a + 49\right)\cdot 61 + \left(40 a^{2} + 53 a + 39\right)\cdot 61^{2} + \left(6 a^{2} + 47 a + 47\right)\cdot 61^{3} + \left(35 a^{2} + 30 a + 7\right)\cdot 61^{4} + \left(12 a^{2} + 27 a + 29\right)\cdot 61^{5} + \left(30 a^{2} + 55 a + 3\right)\cdot 61^{6} + \left(47 a^{2} + 6 a + 39\right)\cdot 61^{7} + \left(60 a^{2} + 3 a + 33\right)\cdot 61^{8} + \left(52 a^{2} + 49 a + 45\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a^{2} + 5 a + 18 + \left(56 a^{2} + 25 a + 53\right)\cdot 61 + \left(9 a^{2} + 27 a + 24\right)\cdot 61^{2} + \left(36 a^{2} + 48 a + 47\right)\cdot 61^{3} + \left(22 a^{2} + a + 35\right)\cdot 61^{4} + \left(11 a^{2} + 36 a + 30\right)\cdot 61^{5} + \left(34 a^{2} + 6 a + 16\right)\cdot 61^{6} + \left(44 a^{2} + 57 a + 13\right)\cdot 61^{7} + \left(23 a^{2} + 33 a + 31\right)\cdot 61^{8} + \left(38 a^{2} + 25 a + 25\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 50 a^{2} + 42 a + 5 + \left(24 a^{2} + 2 a + 23\right)\cdot 61 + \left(5 a^{2} + 47 a + 33\right)\cdot 61^{2} + \left(4 a^{2} + 34 a + 42\right)\cdot 61^{3} + \left(12 a^{2} + 60 a + 17\right)\cdot 61^{4} + \left(25 a^{2} + 43 a + 27\right)\cdot 61^{5} + \left(19 a + 38\right)\cdot 61^{6} + \left(54 a^{2} + 17 a + 39\right)\cdot 61^{7} + \left(13 a^{2} + 49 a + 48\right)\cdot 61^{8} + \left(8 a^{2} + 45 a + 27\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 41 a^{2} + 25 a + 50 + \left(41 a^{2} + a + 46\right)\cdot 61 + \left(7 a^{2} + 26 a + 34\right)\cdot 61^{2} + \left(17 a^{2} + 44 a + 19\right)\cdot 61^{3} + \left(4 a^{2} + 55 a + 11\right)\cdot 61^{4} + \left(20 a^{2} + 58 a + 10\right)\cdot 61^{5} + \left(10 a^{2} + 5 a + 27\right)\cdot 61^{6} + \left(48 a^{2} + 13 a + 50\right)\cdot 61^{7} + \left(42 a^{2} + 60 a + 38\right)\cdot 61^{8} + \left(20 a^{2} + 15 a + 44\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 4 a^{2} + 4 a + 43 + \left(39 a^{2} + 47 a + 56\right)\cdot 61 + \left(39 a^{2} + 38 a + 55\right)\cdot 61^{2} + \left(7 a^{2} + 51 a + 31\right)\cdot 61^{3} + \left(8 a^{2} + 9 a + 44\right)\cdot 61^{4} + \left(12 a^{2} + 6\right)\cdot 61^{5} + \left(8 a^{2} + a + 43\right)\cdot 61^{6} + \left(49 a^{2} + 43 a + 46\right)\cdot 61^{7} + \left(3 a^{2} + 28 a + 11\right)\cdot 61^{8} + \left(7 a^{2} + 12 a + 14\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 4 a^{2} + 9 a + 14 + \left(16 a^{2} + 40 a + 43\right)\cdot 61 + \left(9 a^{2} + 11 a + 10\right)\cdot 61^{2} + \left(43 a^{2} + 38 a + 1\right)\cdot 61^{3} + \left(27 a^{2} + 24 a + 50\right)\cdot 61^{4} + \left(44 a^{2} + 28 a + 35\right)\cdot 61^{5} + \left(15 a^{2} + 32 a + 28\right)\cdot 61^{6} + \left(42 a^{2} + 37 a + 25\right)\cdot 61^{7} + \left(2 a^{2} + 48 a + 57\right)\cdot 61^{8} + \left(55 a^{2} + 8 a + 42\right)\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$12$
$9$$2$$(1,5)$$0$
$18$$2$$(1,2)(5,7)(8,9)$$0$
$27$$2$$(1,5)(2,7)(3,4)$$0$
$27$$2$$(1,5)(2,7)$$-4$
$54$$2$$(1,5)(2,3)(4,7)(6,8)$$0$
$6$$3$$(3,4,6)$$6$
$8$$3$$(1,5,9)(2,7,8)(3,4,6)$$-6$
$12$$3$$(2,7,8)(3,4,6)$$0$
$72$$3$$(1,2,3)(4,5,7)(6,9,8)$$0$
$54$$4$$(1,7,5,2)(8,9)$$0$
$162$$4$$(1,4,5,3)(6,9)(7,8)$$0$
$36$$6$$(1,2)(3,4,6)(5,7)(8,9)$$0$
$36$$6$$(1,3,5,4,9,6)$$0$
$36$$6$$(1,5)(3,4,6)$$0$
$36$$6$$(1,5)(2,7,8)(3,4,6)$$0$
$54$$6$$(1,5)(2,7)(3,6,4)$$2$
$72$$6$$(1,2,5,7,9,8)(3,4,6)$$0$
$108$$6$$(1,5)(2,3,7,4,8,6)$$0$
$216$$6$$(1,7,4,5,2,3)(6,9,8)$$0$
$144$$9$$(1,2,3,5,7,4,9,8,6)$$0$
$108$$12$$(1,7,5,2)(3,4,6)(8,9)$$0$

The blue line marks the conjugacy class containing complex conjugation.