Properties

Label 2.3332.6t3.e.a
Dimension $2$
Group $D_{6}$
Conductor $3332$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(3332\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.44408896.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.68.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.3332.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 9x^{4} - 14x^{3} + 90x^{2} - 112x + 64 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.

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

Roots:
$r_{ 1 }$ $=$ \( 14 a + 19 + \left(25 a + 19\right)\cdot 29 + \left(6 a + 3\right)\cdot 29^{2} + \left(2 a + 18\right)\cdot 29^{3} + \left(7 a + 14\right)\cdot 29^{4} + \left(23 a + 15\right)\cdot 29^{5} + \left(16 a + 14\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 2 + \left(3 a + 17\right)\cdot 29 + \left(22 a + 12\right)\cdot 29^{2} + \left(26 a + 22\right)\cdot 29^{3} + \left(21 a + 18\right)\cdot 29^{4} + \left(5 a + 8\right)\cdot 29^{5} + \left(12 a + 17\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 25 + 19\cdot 29 + 16\cdot 29^{3} + 6\cdot 29^{4} + 17\cdot 29^{5} + 10\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 23 + 9\cdot 29^{2} + 23\cdot 29^{3} + 23\cdot 29^{4} + 19\cdot 29^{5} + 21\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 25 a + 19 + \left(19 a + 6\right)\cdot 29 + \left(4 a + 14\right)\cdot 29^{2} + \left(4 a + 24\right)\cdot 29^{3} + \left(4 a + 17\right)\cdot 29^{4} + \left(25 a + 24\right)\cdot 29^{5} + \left(22 a + 24\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 28 + \left(9 a + 22\right)\cdot 29 + \left(24 a + 17\right)\cdot 29^{2} + \left(24 a + 11\right)\cdot 29^{3} + \left(24 a + 5\right)\cdot 29^{4} + \left(3 a + 1\right)\cdot 29^{5} + \left(6 a + 27\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.