Properties

Label 4.6008.8t44.d.a
Dimension $4$
Group $C_2 \wr S_4$
Conductor $6008$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_2 \wr S_4$
Conductor: \(6008\)\(\medspace = 2^{3} \cdot 751 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.4512008.1
Galois orbit size: $1$
Smallest permutation container: $C_2 \wr S_4$
Parity: odd
Determinant: 1.6008.2t1.a.a
Projective image: $C_2^3:S_4$
Projective stem field: Galois closure of 8.4.2310148096.2

Defining polynomial

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.