Properties

Label 2.168.6t5.c.b
Dimension $2$
Group $S_3\times C_3$
Conductor $168$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Artin stem field: Galois closure of 6.0.677376.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.168.6t1.c.b
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.1176.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 8x^{2} - 4x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.