Properties

Label 4.14414517.6t10.e.a
Dimension $4$
Group $C_3^2:C_4$
Conductor $14414517$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:C_4$
Conductor: \(14414517\)\(\medspace = 3^{8} \cdot 13^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.2313441.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:C_4$
Parity: even
Determinant: 1.13.2t1.a.a
Projective image: $C_3^2:C_4$
Projective stem field: Galois closure of 6.2.2313441.1

Defining polynomial

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.