# Properties

 Label 2.17545.4t3.b.a Dimension 2 Group $D_{4}$ Conductor $5 \cdot 11^{2} \cdot 29$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $17545= 5 \cdot 11^{2} \cdot 29$ Artin number field: Splitting field of 4.0.87725.1 defined by $f= x^{4} - x^{3} + 30 x^{2} - 8 x + 229$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Even Determinant: 1.145.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{29})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $13 + 73\cdot 151 + 126\cdot 151^{2} + 149\cdot 151^{3} + 56\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 2 }$ $=$ $34 + 97\cdot 151 + 83\cdot 151^{2} + 127\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 3 }$ $=$ $111 + 132\cdot 151 + 10\cdot 151^{2} + 97\cdot 151^{3} + 135\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 4 }$ $=$ $145 + 149\cdot 151 + 80\cdot 151^{2} + 78\cdot 151^{3} + 84\cdot 151^{4} +O\left(151^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,4)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,3)$ $0$ $2$ $4$ $(1,4,3,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.