# Properties

 Label 2.323.4t3.a.a Dimension 2 Group $D_{4}$ Conductor $17 \cdot 19$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $323= 17 \cdot 19$ Artin number field: Splitting field of 4.2.5491.1 defined by $f= x^{4} - 2 x^{3} - 2 x^{2} + 3 x - 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.323.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{17}, \sqrt{-19})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $20 + 79\cdot 83 + 33\cdot 83^{2} + 81\cdot 83^{3} + 5\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 2 }$ $=$ $27 + 81\cdot 83 + 41\cdot 83^{2} + 81\cdot 83^{3} + 30\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 3 }$ $=$ $57 + 83 + 41\cdot 83^{2} + 83^{3} + 52\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 4 }$ $=$ $64 + 3\cdot 83 + 49\cdot 83^{2} + 83^{3} + 77\cdot 83^{4} +O\left(83^{ 5 }\right)$

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

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

### 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.