Properties

Label 2.44.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $44$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(44\)\(\medspace = 2^{2} \cdot 11 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: 6.0.21296.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.11.2t1.a.a
Projective image: $S_3$
Projective field: 6.0.21296.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 2 x^{4} - 3 x^{3} + 2 x^{2} - x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \(x^{2} + 6 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 1 + \left(4 a + 6\right)\cdot 7 + 5 a\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 4 + \left(4 a + 3\right)\cdot 7 + \left(2 a + 2\right)\cdot 7^{2} + \left(5 a + 5\right)\cdot 7^{3} + \left(a + 4\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 3 + 6\cdot 7 + 3\cdot 7^{2} + 3 a\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 2 + \left(2 a + 2\right)\cdot 7 + \left(a + 2\right)\cdot 7^{2} + 4 a\cdot 7^{3} + \left(3 a + 5\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + 6 + \left(2 a + 5\right)\cdot 7 + 4 a\cdot 7^{2} + \left(a + 1\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} +O(7^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 6 + \left(6 a + 3\right)\cdot 7 + \left(6 a + 3\right)\cdot 7^{2} + \left(3 a + 3\right)\cdot 7^{3} + 4\cdot 7^{4} +O(7^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$3$$2$$(1,4)(2,5)(3,6)$$0$
$2$$3$$(1,2,3)(4,6,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.