Properties

Label 1.171.6t1.a.a
Dimension $1$
Group $C_6$
Conductor $171$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(171\)\(\medspace = 3^{2} \cdot 19 \)
Artin field: 6.6.135005697.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{171}(56,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 30 x^{4} - 14 x^{3} + 225 x^{2} + 210 x - 179\)  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:
$r_{ 1 }$ $=$ \( 2 a + 6 + \left(2 a + 2\right)\cdot 17 + \left(5 a + 8\right)\cdot 17^{2} + \left(16 a + 12\right)\cdot 17^{3} + \left(11 a + 16\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + 11 + \left(15 a + 8\right)\cdot 17 + \left(5 a + 6\right)\cdot 17^{2} + \left(14 a + 7\right)\cdot 17^{3} + \left(12 a + 1\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 9 + \left(13 a + 15\right)\cdot 17 + 11\cdot 17^{2} + \left(15 a + 5\right)\cdot 17^{3} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 15 a + 8 + \left(14 a + 2\right)\cdot 17 + \left(11 a + 11\right)\cdot 17^{2} + 6\cdot 17^{3} + \left(5 a + 12\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 15 + \left(3 a + 5\right)\cdot 17 + \left(16 a + 16\right)\cdot 17^{2} + \left(a + 2\right)\cdot 17^{3} + \left(16 a + 3\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 9 a + 2 + \left(a + 16\right)\cdot 17 + \left(11 a + 13\right)\cdot 17^{2} + \left(2 a + 15\right)\cdot 17^{3} + \left(4 a + 16\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.