Properties

Label 3.8183.6t6.a.a
Dimension $3$
Group $A_4\times C_2$
Conductor $8183$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4\times C_2$
Conductor: \(8183\)\(\medspace = 7^{2} \cdot 167 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.400967.1
Galois orbit size: $1$
Smallest permutation container: $A_4\times C_2$
Parity: odd
Determinant: 1.167.2t1.a.a
Projective image: $A_4$
Projective stem field: Galois closure of 4.4.1366561.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 5x^{4} - 7x^{3} + 10x^{2} - 8x + 8 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.