Properties

Label 2.3744.6t3.d.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: Galois closure of 6.0.9345024.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.104.2t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.104.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 7x^{4} - 16x^{3} + 25x^{2} - 42x + 31 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 24 a + 19 + \left(20 a + 6\right)\cdot 29 + \left(20 a + 16\right)\cdot 29^{2} + \left(7 a + 22\right)\cdot 29^{3} + \left(10 a + 23\right)\cdot 29^{4} + \left(27 a + 17\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 + 2\cdot 29 + 8\cdot 29^{2} + 27\cdot 29^{3} + 2\cdot 29^{4} + 6\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 a + 16 + \left(7 a + 17\right)\cdot 29 + \left(27 a + 18\right)\cdot 29^{2} + \left(19 a + 22\right)\cdot 29^{3} + \left(15 a + 12\right)\cdot 29^{4} + \left(28 a + 20\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 a + 23 + \left(8 a + 28\right)\cdot 29 + \left(8 a + 11\right)\cdot 29^{2} + \left(21 a + 11\right)\cdot 29^{3} + \left(18 a + 9\right)\cdot 29^{4} + \left(a + 28\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 19 + \left(21 a + 8\right)\cdot 29 + \left(a + 2\right)\cdot 29^{2} + \left(9 a + 8\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 17 + 22\cdot 29 + 24\cdot 29^{3} + 24\cdot 29^{4} + 11\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.