Properties

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

Related objects

Learn more

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(171\)\(\medspace = 3^{2} \cdot 19 \)
Artin field: 6.6.48737056617.2
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{171}(122,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 57 x^{4} - 76 x^{3} + 684 x^{2} + 1824 x + 1216\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.