Properties

Label 2.2352.6t3.d
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 6.2.38723328.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.588.1

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 $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 6 a + 20 + \left(19 a + 12\right)\cdot 29 + \left(6 a + 27\right)\cdot 29^{2} + \left(14 a + 13\right)\cdot 29^{3} + 7\cdot 29^{4} + \left(27 a + 6\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 28\cdot 29 + 10\cdot 29^{2} + 5\cdot 29^{3} + 3\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 23 a + 21 + \left(9 a + 15\right)\cdot 29 + \left(22 a + 12\right)\cdot 29^{2} + \left(14 a + 20\right)\cdot 29^{3} + \left(28 a + 24\right)\cdot 29^{4} + \left(a + 24\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 23 a + 9 + \left(9 a + 16\right)\cdot 29 + \left(22 a + 1\right)\cdot 29^{2} + \left(14 a + 15\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} + \left(a + 22\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 + 18\cdot 29^{2} + 23\cdot 29^{3} + 25\cdot 29^{4} + 26\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 6 a + 8 + \left(19 a + 13\right)\cdot 29 + \left(6 a + 16\right)\cdot 29^{2} + \left(14 a + 8\right)\cdot 29^{3} + 4\cdot 29^{4} + \left(27 a + 4\right)\cdot 29^{5} +O(29^{6})\) Copy content 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.