Properties

Label 12.198...088.18t315.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $1.987\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(19871009045337088\)\(\medspace = 2^{10} \cdot 23^{4} \cdot 37^{5}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.11035059968.1
Galois orbit size: $1$
Smallest permutation container: 18T315
Parity: even
Determinant: 1.37.2t1.a.a
Projective image: $C_3^3.S_4.C_2$
Projective stem field: Galois closure of 9.3.11035059968.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 2x^{7} - 2x^{6} + 6x^{5} - 2x^{4} + 2x^{3} - 6x^{2} - 3x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{3} + 2x + 68 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 36 a^{2} + 16 a + 40 + \left(58 a^{2} + 39 a + 33\right)\cdot 73 + \left(7 a^{2} + 20 a + 13\right)\cdot 73^{2} + \left(6 a^{2} + 39 a + 9\right)\cdot 73^{3} + \left(44 a^{2} + 21 a + 4\right)\cdot 73^{4} + \left(70 a^{2} + 31 a + 58\right)\cdot 73^{5} + \left(40 a^{2} + 34 a + 11\right)\cdot 73^{6} + \left(54 a^{2} + 24 a + 24\right)\cdot 73^{7} + \left(7 a^{2} + 38 a + 44\right)\cdot 73^{8} + \left(30 a^{2} + 58 a + 17\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 65 a^{2} + 15 a + 30 + \left(45 a^{2} + 46 a + 65\right)\cdot 73 + \left(22 a^{2} + 58 a + 8\right)\cdot 73^{2} + \left(8 a^{2} + 22 a + 12\right)\cdot 73^{3} + \left(51 a^{2} + 71 a + 62\right)\cdot 73^{4} + \left(39 a^{2} + 40 a + 16\right)\cdot 73^{5} + \left(35 a^{2} + 67 a + 53\right)\cdot 73^{6} + \left(42 a^{2} + 20 a + 56\right)\cdot 73^{7} + \left(45 a^{2} + 54 a + 21\right)\cdot 73^{8} + \left(21 a^{2} + 26 a + 6\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 27 a^{2} + 48 a + 59 + \left(34 a^{2} + 62 a\right)\cdot 73 + \left(66 a^{2} + 28 a + 8\right)\cdot 73^{2} + \left(17 a^{2} + 17 a + 72\right)\cdot 73^{3} + \left(4 a^{2} + 50 a + 66\right)\cdot 73^{4} + \left(66 a^{2} + 28 a + 16\right)\cdot 73^{5} + \left(7 a^{2} + 13 a + 37\right)\cdot 73^{6} + \left(60 a^{2} + 39 a + 16\right)\cdot 73^{7} + \left(34 a + 42\right)\cdot 73^{8} + \left(25 a^{2} + 23 a + 38\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 a^{2} + 22 + \left(36 a^{2} + 11 a + 16\right)\cdot 73 + \left(20 a^{2} + 15 a + 56\right)\cdot 73^{2} + \left(61 a^{2} + 53 a + 56\right)\cdot 73^{3} + \left(13 a^{2} + 37 a + 11\right)\cdot 73^{4} + \left(63 a^{2} + 51 a + 21\right)\cdot 73^{5} + \left(55 a^{2} + 38 a + 66\right)\cdot 73^{6} + \left(6 a^{2} + 13 a + 23\right)\cdot 73^{7} + \left(24 a^{2} + 37 a + 54\right)\cdot 73^{8} + \left(11 a^{2} + 36 a + 56\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 63 a^{2} + 34 + \left(60 a^{2} + 29 a + 60\right)\cdot 73 + \left(4 a^{2} + 56 a + 71\right)\cdot 73^{2} + \left(72 a^{2} + 51 a + 46\right)\cdot 73^{3} + \left(46 a^{2} + 7 a + 26\right)\cdot 73^{4} + \left(38 a^{2} + 72 a + 53\right)\cdot 73^{5} + \left(27 a^{2} + 61 a + 14\right)\cdot 73^{6} + \left(29 a^{2} + 10 a + 24\right)\cdot 73^{7} + \left(59 a^{2} + 37 a + 47\right)\cdot 73^{8} + \left(30 a^{2} + 37 a + 70\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a^{2} + 42 a + 18 + \left(62 a^{2} + 53 a + 2\right)\cdot 73 + \left(66 a^{2} + 35 a + 45\right)\cdot 73^{2} + \left(68 a^{2} + 2 a + 42\right)\cdot 73^{3} + \left(16 a^{2} + 57 a + 64\right)\cdot 73^{4} + \left(61 a^{2} + 70 a + 42\right)\cdot 73^{5} + \left(28 a^{2} + 68 a + 54\right)\cdot 73^{6} + \left(15 a^{2} + 23 a + 59\right)\cdot 73^{7} + \left(59 a^{2} + 49 a + 3\right)\cdot 73^{8} + \left(34 a^{2} + 71 a + 15\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 22 a^{2} + 31 a + 64 + \left(47 a^{2} + 8 a + 30\right)\cdot 73 + \left(58 a^{2} + 22 a + 58\right)\cdot 73^{2} + \left(15 a^{2} + 17 a + 44\right)\cdot 73^{3} + \left(42 a^{2} + 51 a + 49\right)\cdot 73^{4} + \left(21 a^{2} + 23 a + 38\right)\cdot 73^{5} + \left(61 a^{2} + 38 a\right)\cdot 73^{6} + \left(50 a^{2} + 35 a + 34\right)\cdot 73^{7} + \left(62 a^{2} + 59 a + 8\right)\cdot 73^{8} + \left(26 a^{2} + 37 a + 53\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 56 a^{2} + 25 a + 49 + \left(50 a^{2} + 54 a + 22\right)\cdot 73 + \left(a^{2} + 60 a + 43\right)\cdot 73^{2} + \left(56 a^{2} + 3 a + 25\right)\cdot 73^{3} + \left(21 a^{2} + 15 a + 17\right)\cdot 73^{4} + \left(41 a^{2} + 45 a + 8\right)\cdot 73^{5} + \left(37 a^{2} + 70 a + 28\right)\cdot 73^{6} + \left(56 a^{2} + 22 a + 60\right)\cdot 73^{7} + \left(12 a^{2} + a + 33\right)\cdot 73^{8} + \left(17 a^{2} + 12 a + 52\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 45 a^{2} + 42 a + 52 + \left(41 a^{2} + 60 a + 59\right)\cdot 73 + \left(42 a^{2} + 66 a + 59\right)\cdot 73^{2} + \left(58 a^{2} + 10 a + 54\right)\cdot 73^{3} + \left(50 a^{2} + 53 a + 61\right)\cdot 73^{4} + \left(35 a^{2} + 35\right)\cdot 73^{5} + \left(69 a^{2} + 44 a + 25\right)\cdot 73^{6} + \left(48 a^{2} + 27 a + 65\right)\cdot 73^{7} + \left(19 a^{2} + 53 a + 35\right)\cdot 73^{8} + \left(21 a^{2} + 60 a + 54\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.