Properties

Label 2.243.6t5.b.b
Dimension $2$
Group $S_3\times C_3$
Conductor $243$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(243\)\(\medspace = 3^{5}\)
Artin stem field: Galois closure of 6.0.177147.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.9.6t1.a.b
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.243.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.