Properties

Label 2.2e3_13.6t3.1c1
Dimension 2
Group $D_{6}$
Conductor $ 2^{3} \cdot 13 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$104= 2^{3} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} + 2 x^{4} - 2 x^{3} + 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd
Determinant: 1.2e3_13.2t1.2c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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