Properties

Label 2.3639.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $3639$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(3639\)\(\medspace = 3 \cdot 1213 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.39726963.2
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.3639.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.3639.1

Defining polynomial

$f(x)$$=$\(x^{6} - 17 x^{4} + 52 x^{2} + 192\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \(x^{2} + 33 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a + 28 + \left(8 a + 19\right)\cdot 37 + \left(8 a + 8\right)\cdot 37^{2} + \left(8 a + 21\right)\cdot 37^{3} + \left(4 a + 3\right)\cdot 37^{4} + \left(24 a + 31\right)\cdot 37^{5} + \left(6 a + 13\right)\cdot 37^{6} +O(37^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 22 + \left(8 a + 27\right)\cdot 37 + \left(8 a + 3\right)\cdot 37^{2} + \left(8 a + 28\right)\cdot 37^{3} + \left(4 a + 24\right)\cdot 37^{4} + \left(24 a + 24\right)\cdot 37^{5} + \left(6 a + 20\right)\cdot 37^{6} +O(37^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 13\cdot 37 + 11\cdot 37^{2} + 23\cdot 37^{3} + 22\cdot 37^{4} + 34\cdot 37^{5} + 18\cdot 37^{6} +O(37^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 31 a + 9 + \left(28 a + 17\right)\cdot 37 + \left(28 a + 28\right)\cdot 37^{2} + \left(28 a + 15\right)\cdot 37^{3} + \left(32 a + 33\right)\cdot 37^{4} + \left(12 a + 5\right)\cdot 37^{5} + \left(30 a + 23\right)\cdot 37^{6} +O(37^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 31 a + 15 + \left(28 a + 9\right)\cdot 37 + \left(28 a + 33\right)\cdot 37^{2} + \left(28 a + 8\right)\cdot 37^{3} + \left(32 a + 12\right)\cdot 37^{4} + \left(12 a + 12\right)\cdot 37^{5} + \left(30 a + 16\right)\cdot 37^{6} +O(37^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 22 + 23\cdot 37 + 25\cdot 37^{2} + 13\cdot 37^{3} + 14\cdot 37^{4} + 2\cdot 37^{5} + 18\cdot 37^{6} +O(37^{7})\)  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.