Properties

Label 18.508...784.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $5.085\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(508\!\cdots\!784\)\(\medspace = 2^{36} \cdot 3^{22} \cdot 11^{9}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.4.32788343808.3
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: even
Determinant: 1.44.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.4.32788343808.3

Defining polynomial

$f(x)$$=$ \( x^{8} + 3x^{6} - 14x^{5} - 18x^{4} + 12x^{3} + 31x^{2} + 18x + 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 20\cdot 29 + 8\cdot 29^{2} + 14\cdot 29^{3} + 13\cdot 29^{4} + 21\cdot 29^{5} + 4\cdot 29^{6} + 5\cdot 29^{7} + 5\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 24 + \left(16 a + 13\right)\cdot 29 + 5\cdot 29^{2} + \left(22 a + 1\right)\cdot 29^{3} + \left(26 a + 20\right)\cdot 29^{4} + \left(20 a + 3\right)\cdot 29^{5} + \left(22 a + 6\right)\cdot 29^{6} + \left(16 a + 25\right)\cdot 29^{7} + \left(5 a + 6\right)\cdot 29^{8} + \left(28 a + 22\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 11\cdot 29 + 20\cdot 29^{2} + 17\cdot 29^{3} + 18\cdot 29^{4} + 6\cdot 29^{5} + 3\cdot 29^{6} + 23\cdot 29^{7} + 23\cdot 29^{8} + 3\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a + 4 + \left(13 a + 13\right)\cdot 29 + \left(10 a + 9\right)\cdot 29^{2} + \left(19 a + 13\right)\cdot 29^{3} + \left(15 a + 12\right)\cdot 29^{4} + \left(21 a + 12\right)\cdot 29^{5} + \left(14 a + 13\right)\cdot 29^{6} + 6\cdot 29^{7} + \left(24 a + 27\right)\cdot 29^{8} + \left(27 a + 17\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 21 a + 15 + \left(15 a + 13\right)\cdot 29 + \left(18 a + 19\right)\cdot 29^{2} + \left(9 a + 12\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} + \left(7 a + 17\right)\cdot 29^{5} + \left(14 a + 7\right)\cdot 29^{6} + \left(28 a + 23\right)\cdot 29^{7} + \left(4 a + 1\right)\cdot 29^{8} + \left(a + 17\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 5 a + 13 + \left(4 a + 2\right)\cdot 29 + \left(8 a + 12\right)\cdot 29^{2} + 2 a\cdot 29^{3} + \left(10 a + 16\right)\cdot 29^{4} + \left(8 a + 28\right)\cdot 29^{5} + \left(18 a + 21\right)\cdot 29^{6} + \left(14 a + 3\right)\cdot 29^{7} + \left(21 a + 28\right)\cdot 29^{8} + \left(10 a + 9\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 24 a + 9 + \left(24 a + 18\right)\cdot 29 + \left(20 a + 19\right)\cdot 29^{2} + \left(26 a + 3\right)\cdot 29^{3} + \left(18 a + 6\right)\cdot 29^{4} + \left(20 a + 2\right)\cdot 29^{5} + \left(10 a + 18\right)\cdot 29^{6} + 14 a\cdot 29^{7} + \left(7 a + 5\right)\cdot 29^{8} + \left(18 a + 13\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 14 a + 12 + \left(12 a + 23\right)\cdot 29 + \left(28 a + 20\right)\cdot 29^{2} + \left(6 a + 23\right)\cdot 29^{3} + \left(2 a + 15\right)\cdot 29^{4} + \left(8 a + 23\right)\cdot 29^{5} + \left(6 a + 11\right)\cdot 29^{6} + \left(12 a + 28\right)\cdot 29^{7} + \left(23 a + 17\right)\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.