# Properties

 Label 2.1152.4t3.c.a Dimension $2$ Group $D_{4}$ Conductor $1152$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$1152$$$$\medspace = 2^{7} \cdot 3^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 4.0.4608.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.8.2t1.b.a Projective image: $C_2^2$ Projective field: $$\Q(\zeta_{8})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 6 x^{2} + 18$$  .

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$5 + 6\cdot 17 + 10\cdot 17^{2} + 11\cdot 17^{3} + 4\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$7 + 10\cdot 17 + 9\cdot 17^{2} + 8\cdot 17^{3} + 6\cdot 17^{4} +O(17^{5})$$ $r_{ 3 }$ $=$ $$10 + 6\cdot 17 + 7\cdot 17^{2} + 8\cdot 17^{3} + 10\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$12 + 10\cdot 17 + 6\cdot 17^{2} + 5\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.