Properties

Label 4.2e6_281.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{6} \cdot 281 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$17984= 2^{6} \cdot 281 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.281.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 14 a + 4 + \left(15 a + 4\right)\cdot 17 + \left(5 a + 1\right)\cdot 17^{2} + \left(14 a + 14\right)\cdot 17^{3} + \left(7 a + 2\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 1 + \left(a + 6\right)\cdot 17 + \left(11 a + 8\right)\cdot 17^{2} + \left(2 a + 5\right)\cdot 17^{3} + \left(9 a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + \left(4 a + 10\right)\cdot 17 + \left(8 a + 15\right)\cdot 17^{2} + \left(3 a + 3\right)\cdot 17^{3} + \left(3 a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 11 + \left(12 a + 3\right)\cdot 17 + \left(8 a + 2\right)\cdot 17^{2} + \left(13 a + 16\right)\cdot 17^{3} + \left(13 a + 12\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 14 + 4\cdot 17 + 11\cdot 17^{2} + 14\cdot 17^{3} + 16\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 + 5\cdot 17 + 12\cdot 17^{2} + 13\cdot 17^{3} + 8\cdot 17^{4} +O\left(17^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,3)(2,4)(5,6)$$0$
$6$$2$$(2,6)$$2$
$9$$2$$(2,6)(4,5)$$0$
$4$$3$$(1,2,6)$$1$
$4$$3$$(1,2,6)(3,4,5)$$-2$
$18$$4$$(1,3)(2,5,6,4)$$0$
$12$$6$$(1,4,2,5,6,3)$$0$
$12$$6$$(2,6)(3,4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.