Properties

Label 16.138...984.24t2912.a.a
Dimension $16$
Group $S_3\wr S_3$
Conductor $1.380\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $S_3\wr S_3$
Conductor: \(138\!\cdots\!984\)\(\medspace = 2^{42} \cdot 11^{12}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.1319329792.1
Galois orbit size: $1$
Smallest permutation container: 24T2912
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_3^3.S_4.C_2$
Projective stem field: Galois closure of 9.1.1319329792.1

Defining polynomial

$f(x)$$=$ \( x^{9} + 3x^{7} - 2x^{6} + 3x^{5} - 4x^{4} - x^{3} + 2x^{2} - 4x + 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{3} + 6x + 65 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 64 a^{2} + 55 a + 28 + \left(51 a^{2} + 42 a + 32\right)\cdot 67 + \left(66 a^{2} + 63 a + 19\right)\cdot 67^{2} + \left(47 a^{2} + 59 a + 30\right)\cdot 67^{3} + \left(27 a^{2} + 63 a + 13\right)\cdot 67^{4} + \left(2 a^{2} + 2 a + 52\right)\cdot 67^{5} + \left(19 a^{2} + 56 a + 54\right)\cdot 67^{6} + \left(46 a^{2} + 30 a + 55\right)\cdot 67^{7} + \left(43 a^{2} + 34 a + 54\right)\cdot 67^{8} + \left(6 a^{2} + 40 a + 11\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 64 a + 8 + \left(48 a^{2} + 11 a + 12\right)\cdot 67 + \left(19 a^{2} + 7 a + 3\right)\cdot 67^{2} + \left(63 a^{2} + 27 a + 5\right)\cdot 67^{3} + \left(14 a^{2} + 60 a + 36\right)\cdot 67^{4} + \left(20 a^{2} + 22 a + 9\right)\cdot 67^{5} + \left(19 a^{2} + 31\right)\cdot 67^{6} + \left(59 a^{2} + 33 a + 5\right)\cdot 67^{7} + \left(43 a^{2} + 64 a + 40\right)\cdot 67^{8} + \left(18 a^{2} + 47 a + 55\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 59 a^{2} + 31 a + 43 + \left(38 a^{2} + 47 a + 42\right)\cdot 67 + \left(33 a^{2} + 18 a + 58\right)\cdot 67^{2} + \left(66 a^{2} + 44 a + 17\right)\cdot 67^{3} + \left(52 a^{2} + 47 a + 54\right)\cdot 67^{4} + \left(27 a^{2} + 35 a + 39\right)\cdot 67^{5} + \left(54 a^{2} + 4 a + 37\right)\cdot 67^{6} + \left(10 a^{2} + 59 a + 12\right)\cdot 67^{7} + \left(23 a^{2} + 32 a + 24\right)\cdot 67^{8} + \left(50 a^{2} + 44 a + 48\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a^{2} + 41 a + 1 + \left(59 a^{2} + 19 a + 61\right)\cdot 67 + \left(10 a^{2} + 58 a + 63\right)\cdot 67^{2} + \left(34 a^{2} + 38 a + 41\right)\cdot 67^{3} + \left(61 a^{2} + 60 a + 14\right)\cdot 67^{4} + \left(37 a^{2} + 14 a + 60\right)\cdot 67^{5} + \left(4 a^{2} + 65 a + 63\right)\cdot 67^{6} + \left(11 a^{2} + 27 a + 48\right)\cdot 67^{7} + \left(22 a^{2} + 34 a + 35\right)\cdot 67^{8} + \left(33 a^{2} + 57 a + 51\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 a^{2} + 13 a + 5 + \left(48 a^{2} + 13 a + 13\right)\cdot 67 + \left(a^{2} + 5 a + 62\right)\cdot 67^{2} + \left(50 a^{2} + 31 a + 6\right)\cdot 67^{3} + \left(35 a^{2} + 38 a + 63\right)\cdot 67^{4} + \left(45 a^{2} + 57 a + 9\right)\cdot 67^{5} + \left(11 a^{2} + 14 a + 47\right)\cdot 67^{6} + \left(2 a^{2} + 23 a + 34\right)\cdot 67^{7} + \left(3 a^{2} + 25 a + 66\right)\cdot 67^{8} + \left(48 a^{2} + 9 a + 24\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 8 a^{2} + 39 a + 40 + \left(47 a^{2} + 7 a + 8\right)\cdot 67 + \left(13 a^{2} + 41 a + 46\right)\cdot 67^{2} + \left(4 a^{2} + 62 a + 36\right)\cdot 67^{3} + \left(66 a^{2} + 25 a + 39\right)\cdot 67^{4} + \left(18 a^{2} + 8 a + 4\right)\cdot 67^{5} + \left(60 a^{2} + 62 a + 61\right)\cdot 67^{6} + \left(63 a^{2} + 41 a + 23\right)\cdot 67^{7} + \left(66 a^{2} + 36 a + 65\right)\cdot 67^{8} + \left(64 a^{2} + 41 a + 39\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 63 a^{2} + 38 a + 24 + \left(22 a^{2} + 4 a + 50\right)\cdot 67 + \left(56 a^{2} + 12 a + 44\right)\cdot 67^{2} + \left(51 a^{2} + 35 a + 45\right)\cdot 67^{3} + \left(44 a^{2} + 9 a + 14\right)\cdot 67^{4} + \left(26 a^{2} + 49 a + 15\right)\cdot 67^{5} + \left(43 a^{2} + 12 a + 18\right)\cdot 67^{6} + \left(9 a^{2} + 8 a + 43\right)\cdot 67^{7} + \left(a^{2} + 65 a + 18\right)\cdot 67^{8} + \left(27 a^{2} + 35 a + 26\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 35 a^{2} + 6 a + 25 + \left(33 a^{2} + 34 a + 20\right)\cdot 67 + \left(3 a^{2} + 27 a + 2\right)\cdot 67^{2} + \left(17 a^{2} + 28 a + 9\right)\cdot 67^{3} + \left(22 a^{2} + 22 a + 9\right)\cdot 67^{4} + \left(63 a^{2} + 57 a + 14\right)\cdot 67^{5} + \left(42 a^{2} + 32 a + 38\right)\cdot 67^{6} + \left(14 a^{2} + 38 a + 17\right)\cdot 67^{7} + \left(38 a^{2} + 53 a + 6\right)\cdot 67^{8} + \left(16 a^{2} + 11 a + 33\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 2 a^{2} + 48 a + 27 + \left(52 a^{2} + 19 a + 27\right)\cdot 67 + \left(61 a^{2} + 34 a + 34\right)\cdot 67^{2} + \left(66 a^{2} + 7 a + 7\right)\cdot 67^{3} + \left(8 a^{2} + 6 a + 23\right)\cdot 67^{4} + \left(25 a^{2} + 19 a + 62\right)\cdot 67^{5} + \left(12 a^{2} + 19 a + 49\right)\cdot 67^{6} + \left(50 a^{2} + 5 a + 25\right)\cdot 67^{7} + \left(25 a^{2} + 55 a + 23\right)\cdot 67^{8} + \left(2 a^{2} + 45 a + 43\right)\cdot 67^{9} +O(67^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$16$
$9$$2$$(1,6)$$0$
$18$$2$$(1,3)(5,6)(7,9)$$0$
$27$$2$$(1,6)(2,4)(3,5)$$0$
$27$$2$$(1,6)(3,5)$$0$
$54$$2$$(1,6)(2,3)(4,5)(7,8)$$0$
$6$$3$$(2,4,8)$$-8$
$8$$3$$(1,6,9)(2,4,8)(3,5,7)$$-2$
$12$$3$$(1,6,9)(2,4,8)$$4$
$72$$3$$(1,3,2)(4,6,5)(7,8,9)$$-2$
$54$$4$$(1,5,6,3)(7,9)$$0$
$162$$4$$(1,4,6,2)(5,7)(8,9)$$0$
$36$$6$$(1,3)(2,4,8)(5,6)(7,9)$$0$
$36$$6$$(1,2,6,4,9,8)$$0$
$36$$6$$(1,6)(2,4,8)$$0$
$36$$6$$(1,6)(2,4,8)(3,5,7)$$0$
$54$$6$$(1,6)(2,8,4)(3,5)$$0$
$72$$6$$(1,3,6,5,9,7)(2,4,8)$$0$
$108$$6$$(1,6)(2,5,4,7,8,3)$$0$
$216$$6$$(1,5,4,6,3,2)(7,8,9)$$0$
$144$$9$$(1,3,2,6,5,4,9,7,8)$$1$
$108$$12$$(1,5,6,3)(2,4,8)(7,9)$$0$

The blue line marks the conjugacy class containing complex conjugation.