# Properties

 Label 2.2520.4t3.e Dimension $2$ Group $D_{4}$ Conductor $2520$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$2520$$$$\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.60480.2 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-6}, \sqrt{-70})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$4 + 38\cdot 53 + 3\cdot 53^{2} + 18\cdot 53^{3} + 18\cdot 53^{4} +O(53^{5})$$ $r_{ 2 }$ $=$ $$28 + 24\cdot 53 + 15\cdot 53^{2} + 7\cdot 53^{3} + 52\cdot 53^{4} +O(53^{5})$$ $r_{ 3 }$ $=$ $$36 + 12\cdot 53 + 51\cdot 53^{2} + 51\cdot 53^{3} + 16\cdot 53^{4} +O(53^{5})$$ $r_{ 4 }$ $=$ $$40 + 30\cdot 53 + 35\cdot 53^{2} + 28\cdot 53^{3} + 18\cdot 53^{4} +O(53^{5})$$

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