Properties

Label 2.2240.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $2240$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(2240\)\(\medspace = 2^{6} \cdot 5 \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.12544000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.35.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.140.1

Defining polynomial

$f(x)$$=$\(x^{6} + 8 x^{4} + 16 x^{2} + 10\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \(x^{2} + 18 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 14\cdot 19 + 7\cdot 19^{2} + 18\cdot 19^{4} + 4\cdot 19^{5} + 8\cdot 19^{6} + 16\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + \left(3 a + 16\right)\cdot 19 + \left(11 a + 18\right)\cdot 19^{2} + \left(a + 4\right)\cdot 19^{3} + \left(8 a + 15\right)\cdot 19^{4} + \left(2 a + 14\right)\cdot 19^{5} + \left(17 a + 15\right)\cdot 19^{6} + \left(15 a + 8\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 17 a + 2 + \left(15 a + 17\right)\cdot 19 + \left(7 a + 7\right)\cdot 19^{2} + \left(17 a + 14\right)\cdot 19^{3} + \left(10 a + 2\right)\cdot 19^{4} + \left(16 a + 9\right)\cdot 19^{5} + \left(a + 11\right)\cdot 19^{6} + \left(3 a + 7\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 4\cdot 19 + 11\cdot 19^{2} + 18\cdot 19^{3} + 14\cdot 19^{5} + 10\cdot 19^{6} + 2\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + \left(15 a + 3\right)\cdot 19 + 7 a\cdot 19^{2} + \left(17 a + 14\right)\cdot 19^{3} + \left(10 a + 3\right)\cdot 19^{4} + \left(16 a + 4\right)\cdot 19^{5} + \left(a + 3\right)\cdot 19^{6} + \left(3 a + 10\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 2 a + 17 + \left(3 a + 1\right)\cdot 19 + \left(11 a + 11\right)\cdot 19^{2} + \left(a + 4\right)\cdot 19^{3} + \left(8 a + 16\right)\cdot 19^{4} + \left(2 a + 9\right)\cdot 19^{5} + \left(17 a + 7\right)\cdot 19^{6} + \left(15 a + 11\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.