Properties

Label 12.116...161.36t2210.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $1.169\times 10^{22}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(116\!\cdots\!161\)\(\medspace = 3^{14} \cdot 367^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.36035099127.1
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 9.3.36035099127.1

Defining polynomial

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.