Properties

Label 8.163...401.24t708.b.a
Dimension $8$
Group $C_2 \wr S_4$
Conductor $1.636\times 10^{18}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $C_2 \wr S_4$
Conductor: \(1635798918612635401\)\(\medspace = 29^{6} \cdot 229^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.1520789.1
Galois orbit size: $1$
Smallest permutation container: 24T708
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^3:S_4$
Projective stem field: Galois closure of 8.4.37090522921.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a + 22 + \left(7 a + 33\right)\cdot 47 + \left(32 a + 11\right)\cdot 47^{2} + \left(7 a + 12\right)\cdot 47^{3} + \left(15 a + 16\right)\cdot 47^{4} + \left(36 a + 28\right)\cdot 47^{5} + \left(28 a + 22\right)\cdot 47^{6} + \left(39 a + 43\right)\cdot 47^{7} + \left(42 a + 10\right)\cdot 47^{8} + \left(10 a + 36\right)\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 26\cdot 47 + 28\cdot 47^{2} + 26\cdot 47^{3} + 28\cdot 47^{4} + 17\cdot 47^{5} + 47^{6} + 18\cdot 47^{7} + 41\cdot 47^{8} + 8\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 46 a + 24 + \left(44 a + 43\right)\cdot 47 + \left(34 a + 13\right)\cdot 47^{2} + \left(21 a + 11\right)\cdot 47^{3} + \left(14 a + 32\right)\cdot 47^{4} + \left(35 a + 24\right)\cdot 47^{5} + \left(15 a + 30\right)\cdot 47^{6} + \left(21 a + 1\right)\cdot 47^{7} + \left(32 a + 40\right)\cdot 47^{8} + \left(2 a + 2\right)\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 35 a + 21 + \left(6 a + 26\right)\cdot 47 + \left(34 a + 30\right)\cdot 47^{2} + \left(33 a + 42\right)\cdot 47^{3} + \left(43 a + 36\right)\cdot 47^{4} + \left(42 a + 19\right)\cdot 47^{5} + \left(8 a + 19\right)\cdot 47^{6} + \left(25 a + 46\right)\cdot 47^{7} + \left(32 a + 45\right)\cdot 47^{8} + \left(41 a + 22\right)\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 a + 44 + \left(40 a + 4\right)\cdot 47 + \left(12 a + 45\right)\cdot 47^{2} + \left(13 a + 28\right)\cdot 47^{3} + \left(3 a + 43\right)\cdot 47^{4} + \left(4 a + 14\right)\cdot 47^{5} + \left(38 a + 41\right)\cdot 47^{6} + \left(21 a + 40\right)\cdot 47^{7} + \left(14 a + 38\right)\cdot 47^{8} + \left(5 a + 26\right)\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 44 a + 28 + \left(39 a + 44\right)\cdot 47 + \left(14 a + 21\right)\cdot 47^{2} + \left(39 a + 42\right)\cdot 47^{3} + \left(31 a + 38\right)\cdot 47^{4} + \left(10 a + 38\right)\cdot 47^{5} + \left(18 a + 43\right)\cdot 47^{6} + \left(7 a + 46\right)\cdot 47^{7} + \left(4 a + 9\right)\cdot 47^{8} + \left(36 a + 15\right)\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 17 + 15\cdot 47 + 44\cdot 47^{2} + 3\cdot 47^{3} + 46\cdot 47^{4} + 9\cdot 47^{5} + 2\cdot 47^{6} + 9\cdot 47^{7} + 11\cdot 47^{8} + 5\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( a + 22 + \left(2 a + 40\right)\cdot 47 + \left(12 a + 38\right)\cdot 47^{2} + \left(25 a + 19\right)\cdot 47^{3} + \left(32 a + 39\right)\cdot 47^{4} + \left(11 a + 33\right)\cdot 47^{5} + \left(31 a + 26\right)\cdot 47^{6} + \left(25 a + 28\right)\cdot 47^{7} + \left(14 a + 36\right)\cdot 47^{8} + \left(44 a + 22\right)\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$8$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-8$
$4$$2$$(2,7)$$-4$
$4$$2$$(1,8)(2,7)(3,6)$$4$
$6$$2$$(2,7)(4,5)$$0$
$12$$2$$(1,3)(2,4)(5,7)(6,8)$$0$
$12$$2$$(1,4)(5,8)$$0$
$12$$2$$(1,8)(2,3)(4,5)(6,7)$$0$
$24$$2$$(1,4)(2,7)(5,8)$$0$
$32$$3$$(1,3,2)(6,7,8)$$-1$
$12$$4$$(1,3,8,6)(2,5,7,4)$$0$
$12$$4$$(1,4,8,5)$$0$
$12$$4$$(1,8)(2,6,7,3)(4,5)$$0$
$24$$4$$(1,3)(2,5,7,4)(6,8)$$0$
$24$$4$$(1,4,8,5)(2,7)$$0$
$48$$4$$(1,4,3,2)(5,6,7,8)$$0$
$32$$6$$(1,3,2,8,6,7)$$1$
$32$$6$$(1,3,2)(4,5)(6,7,8)$$-1$
$32$$6$$(1,3,2,8,6,7)(4,5)$$1$
$48$$8$$(1,4,3,2,8,5,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.