Properties

Label 4.17856.6t13.a.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $17856$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(17856\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 31 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.4.91517952.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Determinant: 1.124.2t1.a.a
Projective image: $\SOPlus(4,2)$
Projective stem field: Galois closure of 6.4.91517952.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 4x^{4} - 16x^{3} - 27x^{2} + 32x + 64 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 15 a + 27 + \left(3 a + 36\right)\cdot 43 + \left(22 a + 8\right)\cdot 43^{2} + \left(14 a + 39\right)\cdot 43^{3} + \left(9 a + 38\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 18\cdot 43 + 14\cdot 43^{2} + 6\cdot 43^{3} + 2\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 39 a + 37 + \left(35 a + 13\right)\cdot 43 + \left(9 a + 27\right)\cdot 43^{2} + \left(37 a + 4\right)\cdot 43^{3} + \left(24 a + 5\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 33 + \left(7 a + 10\right)\cdot 43 + \left(33 a + 1\right)\cdot 43^{2} + \left(5 a + 32\right)\cdot 43^{3} + \left(18 a + 35\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 + 24\cdot 43 + 6\cdot 43^{2} + 15\cdot 43^{3} + 13\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 a + 42 + \left(39 a + 24\right)\cdot 43 + \left(20 a + 27\right)\cdot 43^{2} + \left(28 a + 31\right)\cdot 43^{3} + \left(33 a + 33\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,5)(4,6)$$2$
$6$$2$$(2,3)$$0$
$9$$2$$(1,5)(2,3)$$0$
$4$$3$$(1,5,6)(2,3,4)$$1$
$4$$3$$(2,3,4)$$-2$
$18$$4$$(1,2,5,3)(4,6)$$0$
$12$$6$$(1,2,5,3,6,4)$$-1$
$12$$6$$(1,5,6)(2,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.