Properties

Label 2.416.6t3.d
Dimension $2$
Group $D_{6}$
Conductor $416$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(416\)\(\medspace = 2^{5} \cdot 13 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.346112.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.104.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 10 + 15\cdot 23 + 18\cdot 23^{2} + 10\cdot 23^{3} + 17\cdot 23^{4} + 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 19\cdot 23 + 19\cdot 23^{2} + 3\cdot 23^{3} + 8\cdot 23^{4} + 6\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 18 + \left(4 a + 16\right)\cdot 23 + \left(21 a + 17\right)\cdot 23^{2} + \left(13 a + 2\right)\cdot 23^{3} + \left(6 a + 3\right)\cdot 23^{4} + \left(16 a + 9\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 19 + \left(18 a + 13\right)\cdot 23 + \left(a + 9\right)\cdot 23^{2} + \left(9 a + 9\right)\cdot 23^{3} + \left(16 a + 2\right)\cdot 23^{4} + \left(6 a + 12\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 20 a + 7 + \left(14 a + 8\right)\cdot 23 + \left(15 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(19 a + 6\right)\cdot 23^{4} + \left(4 a + 13\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 1 + \left(8 a + 18\right)\cdot 23 + \left(7 a + 9\right)\cdot 23^{2} + \left(9 a + 15\right)\cdot 23^{3} + \left(3 a + 8\right)\cdot 23^{4} + \left(18 a + 3\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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