# Properties

 Label 2.308.4t3.c.a Dimension 2 Group $D_{4}$ Conductor $2^{2} \cdot 7 \cdot 11$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $308= 2^{2} \cdot 7 \cdot 11$ Artin number field: Splitting field of 4.0.2156.1 defined by $f= x^{4} - x^{3} + x^{2} + 4 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.308.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-7}, \sqrt{11})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $52 + 20\cdot 107 + 87\cdot 107^{2} + 5\cdot 107^{3} + 4\cdot 107^{4} +O\left(107^{ 5 }\right)$ $r_{ 2 }$ $=$ $73 + 69\cdot 107 + 41\cdot 107^{2} + 62\cdot 107^{3} + 69\cdot 107^{4} +O\left(107^{ 5 }\right)$ $r_{ 3 }$ $=$ $93 + 98\cdot 107 + 50\cdot 107^{2} + 86\cdot 107^{3} + 77\cdot 107^{4} +O\left(107^{ 5 }\right)$ $r_{ 4 }$ $=$ $104 + 24\cdot 107 + 34\cdot 107^{2} + 59\cdot 107^{3} + 62\cdot 107^{4} +O\left(107^{ 5 }\right)$

### 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.