Properties

Label 3.8991.6t6.a.a
Dimension $3$
Group $A_4\times C_2$
Conductor $8991$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4\times C_2$
Conductor: \(8991\)\(\medspace = 3^{5} \cdot 37 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.4.728271.1
Galois orbit size: $1$
Smallest permutation container: $A_4\times C_2$
Parity: odd
Determinant: 1.111.2t1.a.a
Projective image: $A_4$
Projective stem field: Galois closure of 4.0.110889.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} + 3x^{4} - x^{3} - 3x^{2} + 3x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + \left(16 a + 1\right)\cdot 17 + \left(3 a + 6\right)\cdot 17^{2} + \left(4 a + 8\right)\cdot 17^{3} + \left(15 a + 11\right)\cdot 17^{4} + \left(16 a + 7\right)\cdot 17^{5} + \left(3 a + 6\right)\cdot 17^{6} + \left(8 a + 6\right)\cdot 17^{7} + \left(13 a + 14\right)\cdot 17^{8} + \left(16 a + 6\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 a + 1 + 16\cdot 17 + \left(13 a + 10\right)\cdot 17^{2} + \left(12 a + 8\right)\cdot 17^{3} + \left(a + 5\right)\cdot 17^{4} + 9\cdot 17^{5} + \left(13 a + 10\right)\cdot 17^{6} + \left(8 a + 10\right)\cdot 17^{7} + \left(3 a + 2\right)\cdot 17^{8} + 10\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 12 + 9 a\cdot 17 + 9 a\cdot 17^{2} + \left(8 a + 9\right)\cdot 17^{3} + \left(6 a + 9\right)\cdot 17^{4} + 5 a\cdot 17^{5} + \left(4 a + 9\right)\cdot 17^{6} + \left(13 a + 12\right)\cdot 17^{7} + \left(10 a + 9\right)\cdot 17^{8} + \left(12 a + 7\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 6 + \left(7 a + 16\right)\cdot 17 + \left(7 a + 16\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(10 a + 7\right)\cdot 17^{4} + \left(11 a + 16\right)\cdot 17^{5} + \left(12 a + 7\right)\cdot 17^{6} + \left(3 a + 4\right)\cdot 17^{7} + \left(6 a + 7\right)\cdot 17^{8} + \left(4 a + 9\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 + 9\cdot 17^{2} + 16\cdot 17^{3} + 7\cdot 17^{5} + 10\cdot 17^{6} + 12\cdot 17^{7} + 16\cdot 17^{8} + 15\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 6 + 16\cdot 17 + 7\cdot 17^{2} + 16\cdot 17^{4} + 9\cdot 17^{5} + 6\cdot 17^{6} + 4\cdot 17^{7} + 17^{9} +O(17^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.