# Properties

 Label 2.111.4t3.a.a Dimension 2 Group $D_{4}$ Conductor $3 \cdot 37$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $111= 3 \cdot 37$ Artin number field: Splitting field of 4.0.333.1 defined by $f= x^{4} - x^{3} - 2 x^{2} + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.111.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{37})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $45 + 77\cdot 127 + 15\cdot 127^{2} + 79\cdot 127^{3} + 89\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 2 }$ $=$ $50 + 20\cdot 127 + 27\cdot 127^{2} + 21\cdot 127^{3} + 117\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 3 }$ $=$ $58 + 67\cdot 127 + 109\cdot 127^{2} + 95\cdot 127^{3} + 38\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 4 }$ $=$ $102 + 88\cdot 127 + 101\cdot 127^{2} + 57\cdot 127^{3} + 8\cdot 127^{4} +O\left(127^{ 5 }\right)$

### 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.