Properties

Label 2.2352.6t3.a
Dimension $2$
Group $D_{6}$
Conductor $2352$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(2352\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.5531904.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.588.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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