# Properties

 Label 2.3332.4t3.b.a Dimension $2$ Group $D_{4}$ Conductor $3332$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$3332$$$$\medspace = 2^{2} \cdot 7^{2} \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.13328.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.68.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 3x^{2} - 14x + 53$$ x^4 - 3*x^2 - 14*x + 53 .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$3 + 9\cdot 13 + 12\cdot 13^{2} + 3\cdot 13^{3} + 5\cdot 13^{4} + 8\cdot 13^{5} + 13^{6} +O(13^{7})$$ 3 + 9*13 + 12*13^2 + 3*13^3 + 5*13^4 + 8*13^5 + 13^6+O(13^7) $r_{ 2 }$ $=$ $$5 + 11\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 2\cdot 13^{4} + 12\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})$$ 5 + 11*13 + 11*13^2 + 8*13^3 + 2*13^4 + 12*13^5 + 9*13^6+O(13^7) $r_{ 3 }$ $=$ $$6 + 10\cdot 13 + 3\cdot 13^{2} + 9\cdot 13^{3} + 4\cdot 13^{4} + 8\cdot 13^{5} + 4\cdot 13^{6} +O(13^{7})$$ 6 + 10*13 + 3*13^2 + 9*13^3 + 4*13^4 + 8*13^5 + 4*13^6+O(13^7) $r_{ 4 }$ $=$ $$12 + 7\cdot 13 + 10\cdot 13^{2} + 3\cdot 13^{3} + 10\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})$$ 12 + 7*13 + 10*13^2 + 3*13^3 + 10*13^5 + 9*13^6+O(13^7)

## Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,4)$ $-2$ $2$ $2$ $(1,3)(2,4)$ $0$ $2$ $2$ $(1,2)$ $0$ $2$ $4$ $(1,4,2,3)$ $0$

The blue line marks the conjugacy class containing complex conjugation.