# Properties

 Label 2.276.4t3.f.a Dimension 2 Group $D_{4}$ Conductor $2^{2} \cdot 3 \cdot 23$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $276= 2^{2} \cdot 3 \cdot 23$ Artin number field: Splitting field of 4.2.3312.2 defined by $f= x^{4} + x^{2} - 6 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.276.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{3}, \sqrt{-23})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $11 + 43\cdot 71 + 8\cdot 71^{2} + 31\cdot 71^{3} + 60\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 2 }$ $=$ $17 + 24\cdot 71 + 25\cdot 71^{2} + 54\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 3 }$ $=$ $19 + 15\cdot 71 + 58\cdot 71^{2} + 58\cdot 71^{3} + 5\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 4 }$ $=$ $24 + 59\cdot 71 + 49\cdot 71^{2} + 51\cdot 71^{3} + 21\cdot 71^{4} +O\left(71^{ 5 }\right)$

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

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

### 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.