Properties

Label 8.1546691584.12t213.a.a
Dimension $8$
Group $S_3\wr S_3$
Conductor $1546691584$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $S_3\wr S_3$
Conductor: \(1546691584\)\(\medspace = 2^{10} \cdot 1229^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.118805247296.1
Galois orbit size: $1$
Smallest permutation container: 12T213
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.3.118805247296.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 2x^{8} - x^{7} - x^{6} + 19x^{5} - 41x^{4} + 50x^{3} - 35x^{2} + 4x + 8 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{3} + 3x + 86 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a^{2} + 20 a + 68 + \left(11 a^{2} + 42 a\right)\cdot 89 + \left(71 a^{2} + 46 a + 7\right)\cdot 89^{2} + \left(14 a^{2} + 66 a + 5\right)\cdot 89^{3} + \left(60 a^{2} + 21 a + 77\right)\cdot 89^{4} + \left(24 a^{2} + 64 a + 38\right)\cdot 89^{5} + \left(5 a^{2} + 42 a + 74\right)\cdot 89^{6} + \left(7 a^{2} + 60 a + 28\right)\cdot 89^{7} + \left(82 a^{2} + 46 a + 20\right)\cdot 89^{8} + \left(49 a^{2} + 17 a\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 68 a^{2} + 31 a + 2 + \left(13 a^{2} + 82 a + 6\right)\cdot 89 + \left(39 a^{2} + 57 a + 32\right)\cdot 89^{2} + \left(47 a^{2} + 29 a + 70\right)\cdot 89^{3} + \left(3 a^{2} + 45 a + 52\right)\cdot 89^{4} + \left(50 a^{2} + 64 a\right)\cdot 89^{5} + \left(21 a^{2} + 15 a + 18\right)\cdot 89^{6} + \left(72 a^{2} + 46 a + 70\right)\cdot 89^{7} + \left(74 a^{2} + 71 a + 5\right)\cdot 89^{8} + \left(62 a^{2} + 60 a + 26\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 23 a^{2} + 29 a + 8 + \left(56 a^{2} + 6 a + 61\right)\cdot 89 + \left(64 a^{2} + 5 a + 59\right)\cdot 89^{2} + \left(78 a^{2} + 79 a + 70\right)\cdot 89^{3} + \left(83 a + 28\right)\cdot 89^{4} + \left(5 a^{2} + 22 a + 84\right)\cdot 89^{5} + \left(69 a^{2} + 31 a + 62\right)\cdot 89^{6} + \left(52 a^{2} + 38 a + 57\right)\cdot 89^{7} + \left(12 a^{2} + 41 a + 19\right)\cdot 89^{8} + \left(50 a^{2} + 26 a + 42\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 38 a^{2} + 62 a + 41 + \left(60 a^{2} + 35 a + 45\right)\cdot 89 + \left(78 a^{2} + 70 a + 65\right)\cdot 89^{2} + \left(39 a^{2} + 45 a + 42\right)\cdot 89^{3} + \left(39 a^{2} + 29 a + 65\right)\cdot 89^{4} + \left(20 a^{2} + 4 a + 6\right)\cdot 89^{5} + \left(26 a^{2} + 80 a + 34\right)\cdot 89^{6} + \left(54 a^{2} + 9 a + 82\right)\cdot 89^{7} + \left(10 a^{2} + 40 a + 51\right)\cdot 89^{8} + \left(75 a^{2} + 61 a + 70\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 a^{2} + 38 a + 62 + \left(64 a^{2} + 53 a + 17\right)\cdot 89 + \left(67 a^{2} + 73 a\right)\cdot 89^{2} + \left(26 a^{2} + 81 a + 29\right)\cdot 89^{3} + \left(25 a^{2} + 21 a + 7\right)\cdot 89^{4} + \left(14 a^{2} + 49 a + 18\right)\cdot 89^{5} + \left(62 a^{2} + 30 a + 10\right)\cdot 89^{6} + \left(9 a^{2} + 71 a + 34\right)\cdot 89^{7} + \left(21 a^{2} + 59 a + 76\right)\cdot 89^{8} + \left(65 a^{2} + 10 a + 30\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 63 a^{2} + 17 a + 2 + \left(28 a^{2} + 84 a + 71\right)\cdot 89 + \left(75 a^{2} + 20 a + 58\right)\cdot 89^{2} + \left(59 a^{2} + 54 a + 82\right)\cdot 89^{3} + \left(37 a^{2} + 73 a + 61\right)\cdot 89^{4} + \left(84 a^{2} + 51 a + 45\right)\cdot 89^{5} + \left(27 a^{2} + 20 a + 37\right)\cdot 89^{6} + \left(31 a + 63\right)\cdot 89^{7} + \left(33 a^{2} + 55 a + 7\right)\cdot 89^{8} + \left(5 a^{2} + 57 a + 20\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 14 a^{2} + 17 a + 79 + \left(59 a^{2} + 82 a + 66\right)\cdot 89 + \left(3 a^{2} + 19 a + 26\right)\cdot 89^{2} + \left(51 a^{2} + 16 a + 15\right)\cdot 89^{3} + \left(15 a^{2} + 84 a + 58\right)\cdot 89^{4} + \left(47 a^{2} + 82 a + 79\right)\cdot 89^{5} + \left(82 a^{2} + 63 a\right)\cdot 89^{6} + \left(14 a^{2} + 38 a + 71\right)\cdot 89^{7} + \left(87 a^{2} + 35 a + 79\right)\cdot 89^{8} + \left(25 a^{2} + 71 a + 82\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 52 a^{2} + 43 a + 66 + \left(62 a^{2} + 73\right)\cdot 89 + \left(20 a^{2} + 64 a + 60\right)\cdot 89^{2} + \left(48 a^{2} + 82 a + 9\right)\cdot 89^{3} + \left(72 a^{2} + 9 a + 83\right)\cdot 89^{4} + \left(36 a^{2} + 72 a + 58\right)\cdot 89^{5} + \left(26 a^{2} + 82 a + 66\right)\cdot 89^{6} + \left(21 a^{2} + 11 a + 83\right)\cdot 89^{7} + \left(78 a^{2} + 12 a + 61\right)\cdot 89^{8} + \left(12 a^{2} + 80 a + 56\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 77 a^{2} + 10 a + 30 + \left(88 a^{2} + 58 a + 13\right)\cdot 89 + \left(23 a^{2} + 86 a + 45\right)\cdot 89^{2} + \left(78 a^{2} + 77 a + 30\right)\cdot 89^{3} + \left(11 a^{2} + 74 a + 10\right)\cdot 89^{4} + \left(73 a^{2} + 32 a + 23\right)\cdot 89^{5} + \left(34 a^{2} + 77 a + 51\right)\cdot 89^{6} + \left(34 a^{2} + 47 a + 42\right)\cdot 89^{7} + \left(45 a^{2} + 82 a + 32\right)\cdot 89^{8} + \left(8 a^{2} + 58 a + 26\right)\cdot 89^{9} +O(89^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.