# Properties

 Label 2.1400.4t3.c Dimension $2$ Group $D_{4}$ Conductor $1400$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$1400$$$$\medspace = 2^{3} \cdot 5^{2} \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.9800.2 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{2}, \sqrt{-7})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$14 + 11\cdot 113 + 96\cdot 113^{2} + 105\cdot 113^{3} + 87\cdot 113^{4} +O(113^{5})$$ $r_{ 2 }$ $=$ $$29 + 38\cdot 113 + 96\cdot 113^{2} + 16\cdot 113^{3} + 44\cdot 113^{4} +O(113^{5})$$ $r_{ 3 }$ $=$ $$75 + 52\cdot 113 + 102\cdot 113^{2} + 43\cdot 113^{3} + 99\cdot 113^{4} +O(113^{5})$$ $r_{ 4 }$ $=$ $$109 + 10\cdot 113 + 44\cdot 113^{2} + 59\cdot 113^{3} + 107\cdot 113^{4} +O(113^{5})$$

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