Properties

Label 2.1216.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $1216$
Root number $1$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(1216\)\(\medspace = 2^{6} \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.739328.3
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.19.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.76.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{4} + 4 x^{2} - 2\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \(x^{2} + 29 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 23 a + 1 + 28\cdot 31 + \left(a + 11\right)\cdot 31^{2} + \left(10 a + 10\right)\cdot 31^{3} + \left(17 a + 24\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 30\cdot 31 + 27\cdot 31^{2} + 21\cdot 31^{3} + 3\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 8 a + 16 + \left(30 a + 6\right)\cdot 31 + \left(29 a + 13\right)\cdot 31^{2} + \left(20 a + 29\right)\cdot 31^{3} + \left(13 a + 17\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 8 a + 30 + \left(30 a + 2\right)\cdot 31 + \left(29 a + 19\right)\cdot 31^{2} + \left(20 a + 20\right)\cdot 31^{3} + \left(13 a + 6\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 6 + 3\cdot 31^{2} + 9\cdot 31^{3} + 27\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 23 a + 15 + 24\cdot 31 + \left(a + 17\right)\cdot 31^{2} + \left(10 a + 1\right)\cdot 31^{3} + \left(17 a + 13\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display

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

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

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)(3,6)(4,5)$$0$
$3$$2$$(1,3)(4,6)$$0$
$2$$3$$(1,5,3)(2,6,4)$$-1$
$2$$6$$(1,6,5,4,3,2)$$1$

The blue line marks the conjugacy class containing complex conjugation.