# Properties

 Label 2.55.4t3.c Dimension $2$ Group $D_{4}$ Conductor $55$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$55$$$$\medspace = 5 \cdot 11$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.2.275.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-11})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$3 + 27\cdot 59 + 50\cdot 59^{2} + 50\cdot 59^{3} + 9\cdot 59^{4} +O(59^{5})$$ 3 + 27*59 + 50*59^2 + 50*59^3 + 9*59^4+O(59^5) $r_{ 2 }$ $=$ $$8 + 4\cdot 59 + 33\cdot 59^{2} + 33\cdot 59^{3} + 2\cdot 59^{4} +O(59^{5})$$ 8 + 4*59 + 33*59^2 + 33*59^3 + 2*59^4+O(59^5) $r_{ 3 }$ $=$ $$18 + 49\cdot 59 + 51\cdot 59^{2} + 11\cdot 59^{3} + 34\cdot 59^{4} +O(59^{5})$$ 18 + 49*59 + 51*59^2 + 11*59^3 + 34*59^4+O(59^5) $r_{ 4 }$ $=$ $$31 + 37\cdot 59 + 41\cdot 59^{2} + 21\cdot 59^{3} + 12\cdot 59^{4} +O(59^{5})$$ 31 + 37*59 + 41*59^2 + 21*59^3 + 12*59^4+O(59^5)

### 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 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.