Properties

Label 2.2e3_19.6t3.2c1
Dimension 2
Group $D_{6}$
Conductor $ 2^{3} \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$152= 2^{3} \cdot 19 $
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_19.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 18 a + 19 + \left(29 a + 4\right)\cdot 31 + \left(30 a + 25\right)\cdot 31^{2} + \left(24 a + 4\right)\cdot 31^{3} + \left(30 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 19 + \left(3 a + 28\right)\cdot 31 + \left(4 a + 26\right)\cdot 31^{2} + \left(25 a + 2\right)\cdot 31^{3} + \left(18 a + 1\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 24 + \left(a + 14\right)\cdot 31 + 26\cdot 31^{2} + \left(6 a + 23\right)\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 14 + 3\cdot 31 + 6\cdot 31^{2} + 23\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 + 15\cdot 31 + 7\cdot 31^{2} + 20\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 22 a + 6 + \left(27 a + 26\right)\cdot 31 + 26 a\cdot 31^{2} + \left(5 a + 18\right)\cdot 31^{3} + \left(12 a + 13\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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