Properties

Label 2.416.6t3.c
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 6.0.173056.2
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 $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 8 a + 4 + \left(16 a + 4\right)\cdot 29 + \left(a + 17\right)\cdot 29^{2} + \left(4 a + 22\right)\cdot 29^{3} + 13 a\cdot 29^{4} + \left(14 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 21 a + 15 + \left(12 a + 19\right)\cdot 29 + \left(27 a + 8\right)\cdot 29^{2} + \left(24 a + 12\right)\cdot 29^{3} + \left(15 a + 4\right)\cdot 29^{4} + \left(14 a + 4\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 19 + 23\cdot 29 + 25\cdot 29^{2} + 5\cdot 29^{3} + 5\cdot 29^{4} + 7\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 21 a + 25 + \left(12 a + 24\right)\cdot 29 + \left(27 a + 11\right)\cdot 29^{2} + \left(24 a + 6\right)\cdot 29^{3} + \left(15 a + 28\right)\cdot 29^{4} + \left(14 a + 25\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 14 + \left(16 a + 9\right)\cdot 29 + \left(a + 20\right)\cdot 29^{2} + \left(4 a + 16\right)\cdot 29^{3} + \left(13 a + 24\right)\cdot 29^{4} + \left(14 a + 24\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 10 + 5\cdot 29 + 3\cdot 29^{2} + 23\cdot 29^{3} + 23\cdot 29^{4} + 21\cdot 29^{5} +O\left(29^{ 6 }\right)$

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.