Properties

Label 2.1200.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $1200$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.7200000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.300.1

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 $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 19 a + 6 + \left(6 a + 18\right)\cdot 23 + \left(21 a + 6\right)\cdot 23^{2} + \left(8 a + 7\right)\cdot 23^{3} + \left(20 a + 15\right)\cdot 23^{4} + \left(2 a + 10\right)\cdot 23^{5} +O(23^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 14\cdot 23 + 19\cdot 23^{2} + 11\cdot 23^{3} + 6\cdot 23^{4} + 16\cdot 23^{5} +O(23^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 20 a + 8 + \left(11 a + 17\right)\cdot 23 + \left(13 a + 8\right)\cdot 23^{2} + \left(9 a + 3\right)\cdot 23^{3} + \left(9 a + 6\right)\cdot 23^{4} + \left(8 a + 16\right)\cdot 23^{5} +O(23^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 21 + \left(16 a + 12\right)\cdot 23 + \left(a + 19\right)\cdot 23^{2} + \left(14 a + 3\right)\cdot 23^{3} + \left(2 a + 1\right)\cdot 23^{4} + \left(20 a + 19\right)\cdot 23^{5} +O(23^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 2 + \left(11 a + 21\right)\cdot 23 + 9 a\cdot 23^{2} + \left(13 a + 9\right)\cdot 23^{3} + \left(13 a + 15\right)\cdot 23^{4} + 14 a\cdot 23^{5} +O(23^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 14 + 7\cdot 23 + 13\cdot 23^{2} + 10\cdot 23^{3} + 23^{4} + 6\cdot 23^{5} +O(23^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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