Properties

Label 2.180.6t5.b.b
Dimension $2$
Group $S_3\times C_3$
Conductor $180$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Artin stem field: 6.0.648000.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.180.6t1.b.b
Projective image: $S_3$
Projective stem field: 3.1.1620.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} + 7 x^{2} - 4 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.