Properties

Label 2.1840.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $1840$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(1840\)\(\medspace = 2^{4} \cdot 5 \cdot 23 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.3385600.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.115.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.460.1

Defining polynomial

$f(x)$$=$\(x^{6} - 6 x^{3} + 16 x^{2} - 24 x + 18\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 9 a + 11 + \left(a + 9\right)\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + 5 a\cdot 13^{3} + \left(a + 1\right)\cdot 13^{4} + \left(7 a + 1\right)\cdot 13^{5} + \left(7 a + 11\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 4\cdot 13^{2} + 10\cdot 13^{3} + 13^{4} + 5\cdot 13^{5} + 3\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 7 a + 10 + \left(11 a + 12\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + \left(8 a + 8\right)\cdot 13^{3} + \left(9 a + 12\right)\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(9 a + 2\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 12 + 8\cdot 13 + 7\cdot 13^{2} + 5\cdot 13^{3} + 12\cdot 13^{4} + 2\cdot 13^{5} + 2\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 7 + \left(11 a + 2\right)\cdot 13 + \left(9 a + 5\right)\cdot 13^{2} + \left(7 a + 2\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} + \left(5 a + 6\right)\cdot 13^{5} + \left(5 a + 11\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 6 a + 4 + \left(a + 4\right)\cdot 13 + \left(7 a + 6\right)\cdot 13^{2} + \left(4 a + 11\right)\cdot 13^{3} + 3 a\cdot 13^{4} + \left(10 a + 8\right)\cdot 13^{5} + \left(3 a + 8\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,3)(2,4)(5,6)$$-2$
$3$$2$$(2,5)(4,6)$$0$
$3$$2$$(1,3)(2,6)(4,5)$$0$
$2$$3$$(1,2,5)(3,4,6)$$-1$
$2$$6$$(1,4,5,3,2,6)$$1$

The blue line marks the conjugacy class containing complex conjugation.