# Properties

 Label 2.2475.4t3.c Dimension $2$ Group $D_{4}$ Conductor $2475$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$2475$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 11$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.2.12375.1 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:
 $r_{ 1 }$ $=$ $$2 + 42\cdot 59 + 21\cdot 59^{2} + 23\cdot 59^{3} +O(59^{5})$$ 2 + 42*59 + 21*59^2 + 23*59^3+O(59^5) $r_{ 2 }$ $=$ $$24 + 11\cdot 59 + 4\cdot 59^{2} + 22\cdot 59^{3} + 36\cdot 59^{4} +O(59^{5})$$ 24 + 11*59 + 4*59^2 + 22*59^3 + 36*59^4+O(59^5) $r_{ 3 }$ $=$ $$41 + 49\cdot 59 + 13\cdot 59^{2} + 50\cdot 59^{3} + 10\cdot 59^{4} +O(59^{5})$$ 41 + 49*59 + 13*59^2 + 50*59^3 + 10*59^4+O(59^5) $r_{ 4 }$ $=$ $$52 + 14\cdot 59 + 19\cdot 59^{2} + 22\cdot 59^{3} + 11\cdot 59^{4} +O(59^{5})$$ 52 + 14*59 + 19*59^2 + 22*59^3 + 11*59^4+O(59^5)

### 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.