# Properties

 Label 2.1575.4t3.a Dimension 2 Group $D_{4}$ Conductor $3^{2} \cdot 5^{2} \cdot 7$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $1575= 3^{2} \cdot 5^{2} \cdot 7$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 12 x^{2} - 4 x + 31$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-7})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $14 + 25\cdot 37 + 6\cdot 37^{2} + 36\cdot 37^{3} + 30\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $29 + 30\cdot 37 + 21\cdot 37^{2} + 18\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $34 + 32\cdot 37 + 19\cdot 37^{2} + 27\cdot 37^{3} + 6\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $35 + 21\cdot 37 + 25\cdot 37^{2} + 28\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$

### 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 values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,4)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,3)$ $0$ $2$ $4$ $(1,4,3,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.