Properties

Label 18.586...208.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $5.861\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(586\!\cdots\!208\)\(\medspace = 2^{61} \cdot 3^{26}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.3057647616.1
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 8.2.3057647616.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 10x^{6} - 12x^{5} + 9x^{4} - 8x^{2} + 8x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a^{2} + 11 a + 2 + \left(12 a^{2} + 10 a + 14\right)\cdot 17 + \left(9 a^{2} + a + 5\right)\cdot 17^{2} + \left(a^{2} + 9 a + 3\right)\cdot 17^{3} + \left(11 a^{2} + 16 a + 16\right)\cdot 17^{4} + \left(7 a^{2} + 2 a + 11\right)\cdot 17^{5} + \left(5 a^{2} + 5 a + 6\right)\cdot 17^{6} + \left(7 a^{2} + 14 a + 9\right)\cdot 17^{7} + \left(11 a^{2} + 13 a + 10\right)\cdot 17^{8} + \left(7 a^{2} + 15 a + 9\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 13 + \left(12 a^{2} + 12 a\right)\cdot 17 + \left(8 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(8 a^{2} + 15 a + 5\right)\cdot 17^{3} + \left(12 a^{2} + 11 a + 1\right)\cdot 17^{4} + \left(2 a^{2} + 3 a + 12\right)\cdot 17^{5} + \left(3 a^{2} + 2\right)\cdot 17^{6} + \left(15 a^{2} + 4 a + 8\right)\cdot 17^{7} + \left(5 a^{2} + 4 a + 9\right)\cdot 17^{8} + \left(7 a^{2} + 7 a + 13\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 a^{2} + 3 a + 7 + \left(3 a^{2} + 5 a + 6\right)\cdot 17 + \left(16 a^{2} + a + 15\right)\cdot 17^{2} + \left(5 a^{2} + 14 a + 9\right)\cdot 17^{3} + \left(12 a^{2} + 15 a + 12\right)\cdot 17^{4} + 2 a^{2} 17^{5} + 16 a^{2} 17^{6} + \left(10 a^{2} + 6 a + 11\right)\cdot 17^{7} + \left(5 a^{2} + 13 a + 3\right)\cdot 17^{8} + \left(4 a^{2} + 3 a\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 a^{2} + 16 a + 2 + \left(a^{2} + 15 a + 5\right)\cdot 17 + \left(9 a^{2} + 3 a + 16\right)\cdot 17^{2} + \left(2 a^{2} + 4 a + 1\right)\cdot 17^{3} + \left(9 a^{2} + 6 a + 16\right)\cdot 17^{4} + \left(11 a^{2} + 12 a\right)\cdot 17^{5} + \left(14 a^{2} + 16 a + 16\right)\cdot 17^{6} + \left(7 a^{2} + 6 a + 8\right)\cdot 17^{7} + \left(5 a^{2} + 16 a + 3\right)\cdot 17^{8} + \left(5 a^{2} + 5 a + 12\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 14 + 4\cdot 17 + 3\cdot 17^{2} + 16\cdot 17^{3} + 3\cdot 17^{4} + 3\cdot 17^{5} + 15\cdot 17^{6} + 5\cdot 17^{7} + 8\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 11 a^{2} + 4 a + 4 + \left(12 a + 6\right)\cdot 17 + \left(7 a^{2} + 2 a + 15\right)\cdot 17^{2} + \left(5 a^{2} + 6 a + 5\right)\cdot 17^{3} + \left(9 a^{2} + 9 a + 9\right)\cdot 17^{4} + \left(8 a^{2} + 11 a + 12\right)\cdot 17^{5} + \left(2 a + 14\right)\cdot 17^{6} + \left(11 a^{2} + 14 a + 11\right)\cdot 17^{7} + \left(2 a^{2} + 5 a + 4\right)\cdot 17^{8} + \left(7 a^{2} + 2 a + 9\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 12 + 16\cdot 17 + 17^{2} + 10\cdot 17^{3} + 7\cdot 17^{4} + 13\cdot 17^{5} + 7\cdot 17^{6} + 3\cdot 17^{7} + 8\cdot 17^{8} + 3\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 15 a^{2} + 2 a + 1 + \left(3 a^{2} + 11 a + 14\right)\cdot 17 + \left(12 a + 10\right)\cdot 17^{2} + \left(10 a^{2} + a + 14\right)\cdot 17^{3} + \left(13 a^{2} + 8 a\right)\cdot 17^{4} + \left(2 a + 13\right)\cdot 17^{5} + \left(11 a^{2} + 9 a + 4\right)\cdot 17^{6} + \left(15 a^{2} + 5 a + 9\right)\cdot 17^{7} + \left(2 a^{2} + 14 a + 10\right)\cdot 17^{8} + \left(2 a^{2} + 15 a + 11\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.