Properties

Label 2.1620.6t3.h
Dimension $2$
Group $D_{6}$
Conductor $1620$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.31492800.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.1620.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4\cdot 13 + 2\cdot 13^{2} + 2\cdot 13^{3} + 8\cdot 13^{4} + 13^{5} + 13^{6} + 5\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 5 + \left(12 a + 6\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(9 a + 6\right)\cdot 13^{3} + \left(3 a + 5\right)\cdot 13^{4} + \left(9 a + 9\right)\cdot 13^{5} + \left(a + 9\right)\cdot 13^{6} + 4\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 2 + \left(11 a + 9\right)\cdot 13 + \left(3 a + 4\right)\cdot 13^{2} + \left(11 a + 12\right)\cdot 13^{3} + \left(11 a + 11\right)\cdot 13^{4} + \left(8 a + 7\right)\cdot 13^{5} + \left(10 a + 11\right)\cdot 13^{6} + \left(12 a + 8\right)\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( a + 1 + \left(a + 9\right)\cdot 13 + \left(9 a + 9\right)\cdot 13^{2} + \left(a + 6\right)\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} + \left(4 a + 4\right)\cdot 13^{5} + 2 a\cdot 13^{6} + 11\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 10 + 7\cdot 13 + 11\cdot 13^{2} + 6\cdot 13^{3} + 13^{4} + 13^{6} + 6\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 10 a + 8 + 2\cdot 13 + 12\cdot 13^{2} + \left(3 a + 3\right)\cdot 13^{3} + \left(9 a + 12\right)\cdot 13^{4} + \left(3 a + 1\right)\cdot 13^{5} + \left(11 a + 2\right)\cdot 13^{6} + \left(12 a + 3\right)\cdot 13^{7} +O(13^{8})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,5)(2,3)(4,6)$ $-2$
$3$ $2$ $(2,6)(3,4)$ $0$
$3$ $2$ $(1,3)(2,5)(4,6)$ $0$
$2$ $3$ $(1,2,6)(3,4,5)$ $-1$
$2$ $6$ $(1,3,6,5,2,4)$ $1$
The blue line marks the conjugacy class containing complex conjugation.