Properties

Label 2.1216.6t3.d
Dimension $2$
Group $D_{6}$
Conductor $1216$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(1216\)\(\medspace = 2^{6} \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.739328.2
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.76.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \(x^{2} + 38 x + 6\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4 + 10\cdot 41 + 2\cdot 41^{2} + 28\cdot 41^{3} +O(41^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 22 a + 32 + \left(38 a + 8\right)\cdot 41 + \left(9 a + 2\right)\cdot 41^{2} + \left(7 a + 5\right)\cdot 41^{3} + \left(19 a + 4\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 a + 16 + \left(2 a + 20\right)\cdot 41 + \left(31 a + 34\right)\cdot 41^{2} + \left(33 a + 16\right)\cdot 41^{3} + \left(21 a + 13\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 37 + 30\cdot 41 + 38\cdot 41^{2} + 12\cdot 41^{3} + 40\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 9 + \left(2 a + 32\right)\cdot 41 + \left(31 a + 38\right)\cdot 41^{2} + \left(33 a + 35\right)\cdot 41^{3} + \left(21 a + 36\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 22 a + 25 + \left(38 a + 20\right)\cdot 41 + \left(9 a + 6\right)\cdot 41^{2} + \left(7 a + 24\right)\cdot 41^{3} + \left(19 a + 27\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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