Properties

Label 2.3744.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $3744$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(3744\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 13 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.4672512.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.104.2t1.b.a
Projective image: $S_3$
Projective stem field: 3.1.104.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 10 x^{4} - 4 x^{3} + x^{2} - 6 x - 2\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.