Properties

Label 18.417...568.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $4.173\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(417\!\cdots\!568\)\(\medspace = 2^{51} \cdot 3^{32}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.1719926784.1
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.1719926784.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 8x^{6} - 8x^{5} - x^{4} + 20x^{3} - 28x^{2} + 16x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{2} + 102x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 75 a + 18 + \left(22 a + 74\right)\cdot 103 + \left(55 a + 7\right)\cdot 103^{2} + \left(58 a + 50\right)\cdot 103^{3} + \left(84 a + 61\right)\cdot 103^{4} + \left(94 a + 28\right)\cdot 103^{5} + \left(84 a + 1\right)\cdot 103^{6} + \left(18 a + 43\right)\cdot 103^{7} + \left(15 a + 73\right)\cdot 103^{8} + \left(3 a + 89\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 83 + 50\cdot 103 + 60\cdot 103^{2} + 67\cdot 103^{3} + 2\cdot 103^{4} + 76\cdot 103^{5} + 44\cdot 103^{6} + 34\cdot 103^{7} + 74\cdot 103^{8} + 38\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 39 a + 1 + \left(95 a + 1\right)\cdot 103 + \left(22 a + 2\right)\cdot 103^{2} + \left(48 a + 17\right)\cdot 103^{3} + \left(67 a + 26\right)\cdot 103^{4} + \left(16 a + 12\right)\cdot 103^{5} + \left(16 a + 55\right)\cdot 103^{6} + \left(82 a + 20\right)\cdot 103^{7} + \left(47 a + 102\right)\cdot 103^{8} + \left(46 a + 40\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 24 + \left(35 a + 6\right)\cdot 103 + \left(30 a + 29\right)\cdot 103^{2} + \left(13 a + 52\right)\cdot 103^{3} + \left(91 a + 55\right)\cdot 103^{4} + \left(43 a + 54\right)\cdot 103^{5} + \left(70 a + 3\right)\cdot 103^{6} + \left(29 a + 76\right)\cdot 103^{7} + \left(84 a + 86\right)\cdot 103^{8} + \left(44 a + 98\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 92 a + 35 + \left(67 a + 30\right)\cdot 103 + \left(72 a + 24\right)\cdot 103^{2} + \left(89 a + 35\right)\cdot 103^{3} + \left(11 a + 30\right)\cdot 103^{4} + \left(59 a + 7\right)\cdot 103^{5} + \left(32 a + 30\right)\cdot 103^{6} + \left(73 a + 35\right)\cdot 103^{7} + \left(18 a + 38\right)\cdot 103^{8} + \left(58 a + 59\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 19 + 67\cdot 103 + 9\cdot 103^{2} + 94\cdot 103^{3} + 102\cdot 103^{4} + 26\cdot 103^{5} + 25\cdot 103^{6} + 36\cdot 103^{7} + 2\cdot 103^{8} + 70\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 64 a + 40 + \left(7 a + 57\right)\cdot 103 + \left(80 a + 32\right)\cdot 103^{2} + \left(54 a + 42\right)\cdot 103^{3} + \left(35 a + 45\right)\cdot 103^{4} + \left(86 a + 64\right)\cdot 103^{5} + \left(86 a + 54\right)\cdot 103^{6} + \left(20 a + 86\right)\cdot 103^{7} + \left(55 a + 67\right)\cdot 103^{8} + \left(56 a + 39\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 28 a + 93 + \left(80 a + 21\right)\cdot 103 + \left(47 a + 40\right)\cdot 103^{2} + \left(44 a + 53\right)\cdot 103^{3} + \left(18 a + 87\right)\cdot 103^{4} + \left(8 a + 38\right)\cdot 103^{5} + \left(18 a + 94\right)\cdot 103^{6} + \left(84 a + 79\right)\cdot 103^{7} + \left(87 a + 69\right)\cdot 103^{8} + \left(99 a + 77\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$18$
$6$$2$$(1,5)(4,8)$$-6$
$9$$2$$(1,5)(2,6)(3,7)(4,8)$$2$
$12$$2$$(2,3)$$0$
$24$$2$$(1,2)(3,4)(5,6)(7,8)$$0$
$36$$2$$(1,4)(2,3)$$-2$
$36$$2$$(1,5)(2,3)(4,8)$$0$
$16$$3$$(2,6,7)$$0$
$64$$3$$(2,6,7)(4,5,8)$$0$
$12$$4$$(1,4,5,8)$$0$
$36$$4$$(1,4,5,8)(2,3,6,7)$$-2$
$36$$4$$(1,5)(2,3,6,7)(4,8)$$0$
$72$$4$$(1,6,5,2)(3,4,7,8)$$0$
$72$$4$$(1,4,5,8)(2,3)$$2$
$144$$4$$(1,2,4,3)(5,6)(7,8)$$0$
$48$$6$$(1,5)(2,7,6)(4,8)$$0$
$96$$6$$(2,3)(4,8,5)$$0$
$192$$6$$(1,3)(2,4,6,5,7,8)$$0$
$144$$8$$(1,3,4,6,5,7,8,2)$$0$
$96$$12$$(1,4,5,8)(2,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.