Properties

Label 1.171.6t1.e
Dimension $1$
Group $C_6$
Conductor $171$
Indicator $0$

Related objects

Learn more

Basic invariants

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

Galois action

Roots of defining polynomial

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 values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,6)(2,4)(3,5)$ $-1$ $-1$
$1$ $3$ $(1,4,5)(2,3,6)$ $\zeta_{3}$ $-\zeta_{3} - 1$
$1$ $3$ $(1,5,4)(2,6,3)$ $-\zeta_{3} - 1$ $\zeta_{3}$
$1$ $6$ $(1,3,4,6,5,2)$ $\zeta_{3} + 1$ $-\zeta_{3}$
$1$ $6$ $(1,2,5,6,4,3)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.