Properties

Label 2.2240.6t3.d
Dimension $2$
Group $D_{6}$
Conductor $2240$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(2240\)\(\medspace = 2^{6} \cdot 5 \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.2508800.3
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.140.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 21 a + 3 + \left(22 a + 7\right)\cdot 23 + \left(20 a + 1\right)\cdot 23^{2} + \left(4 a + 18\right)\cdot 23^{3} + \left(12 a + 12\right)\cdot 23^{4} + \left(14 a + 21\right)\cdot 23^{5} + \left(13 a + 22\right)\cdot 23^{6} +O(23^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 22 + 8\cdot 23 + \left(2 a + 20\right)\cdot 23^{2} + \left(18 a + 6\right)\cdot 23^{3} + \left(10 a + 9\right)\cdot 23^{4} + \left(8 a + 15\right)\cdot 23^{5} + \left(9 a + 12\right)\cdot 23^{6} +O(23^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 22\cdot 23 + 21\cdot 23^{2} + 15\cdot 23^{3} + 18\cdot 23^{4} + 13\cdot 23^{5} + 13\cdot 23^{6} +O(23^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 2 a + 20 + 15\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(18 a + 4\right)\cdot 23^{3} + \left(10 a + 10\right)\cdot 23^{4} + \left(8 a + 1\right)\cdot 23^{5} + 9 a\cdot 23^{6} +O(23^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 21 a + 1 + \left(22 a + 14\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(4 a + 16\right)\cdot 23^{3} + \left(12 a + 13\right)\cdot 23^{4} + \left(14 a + 7\right)\cdot 23^{5} + \left(13 a + 10\right)\cdot 23^{6} +O(23^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 3 + 23^{2} + 7\cdot 23^{3} + 4\cdot 23^{4} + 9\cdot 23^{5} + 9\cdot 23^{6} +O(23^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

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