Properties

Label 2.2e4_83.6t3.1c1
Dimension 2
Group $D_{6}$
Conductor $ 2^{4} \cdot 83 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$1328= 2^{4} \cdot 83 $
Artin number field: Splitting field of $f= x^{6} - x^{4} - 3 x^{2} + 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd
Determinant: 1.83.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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