Properties

Label 2.3332.6t3.a.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 6.2.310862272.1
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)$$=$ \( x^{6} + 7x^{4} + 63x^{2} - 7 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 a + 2 + \left(12 a + 6\right)\cdot 19 + 7\cdot 19^{2} + \left(2 a + 16\right)\cdot 19^{3} + \left(15 a + 7\right)\cdot 19^{4} + \left(10 a + 3\right)\cdot 19^{5} + \left(2 a + 12\right)\cdot 19^{6} + \left(6 a + 12\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 18 + \left(6 a + 2\right)\cdot 19 + \left(18 a + 14\right)\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(3 a + 1\right)\cdot 19^{4} + \left(8 a + 18\right)\cdot 19^{5} + \left(16 a + 3\right)\cdot 19^{6} + \left(12 a + 16\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 + 10\cdot 19 + 9\cdot 19^{2} + 9\cdot 19^{3} + 19^{4} + 5\cdot 19^{5} + 17\cdot 19^{6} + 4\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 17 + \left(6 a + 12\right)\cdot 19 + \left(18 a + 11\right)\cdot 19^{2} + \left(16 a + 2\right)\cdot 19^{3} + \left(3 a + 11\right)\cdot 19^{4} + \left(8 a + 15\right)\cdot 19^{5} + \left(16 a + 6\right)\cdot 19^{6} + \left(12 a + 6\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a + 1 + \left(12 a + 16\right)\cdot 19 + 4\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(15 a + 17\right)\cdot 19^{4} + 10 a\cdot 19^{5} + \left(2 a + 15\right)\cdot 19^{6} + \left(6 a + 2\right)\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 10 + 8\cdot 19 + 9\cdot 19^{2} + 9\cdot 19^{3} + 17\cdot 19^{4} + 13\cdot 19^{5} + 19^{6} + 14\cdot 19^{7} +O(19^{8})\) Copy content Toggle raw display

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

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

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)(4,5)$$0$
$3$$2$$(1,4)(2,3)(5,6)$$0$
$2$$3$$(1,2,6)(3,4,5)$$-1$
$2$$6$$(1,3,2,4,6,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.