# Properties

 Label 1.176.4t1.a Dimension $1$ Group $C_4$ Conductor $176$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$176$$$$\medspace = 2^{4} \cdot 11$$ Artin number field: Galois closure of 4.4.247808.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $$6 + 10\cdot 23 + 3\cdot 23^{2} + 15\cdot 23^{3} + 22\cdot 23^{4} + 13\cdot 23^{5} +O(23^{6})$$ 6 + 10*23 + 3*23^2 + 15*23^3 + 22*23^4 + 13*23^5+O(23^6) $r_{ 2 }$ $=$ $$10 + 3\cdot 23 + 5\cdot 23^{2} + 17\cdot 23^{3} + 20\cdot 23^{4} + 12\cdot 23^{5} +O(23^{6})$$ 10 + 3*23 + 5*23^2 + 17*23^3 + 20*23^4 + 12*23^5+O(23^6) $r_{ 3 }$ $=$ $$13 + 19\cdot 23 + 17\cdot 23^{2} + 5\cdot 23^{3} + 2\cdot 23^{4} + 10\cdot 23^{5} +O(23^{6})$$ 13 + 19*23 + 17*23^2 + 5*23^3 + 2*23^4 + 10*23^5+O(23^6) $r_{ 4 }$ $=$ $$17 + 12\cdot 23 + 19\cdot 23^{2} + 7\cdot 23^{3} + 9\cdot 23^{5} +O(23^{6})$$ 17 + 12*23 + 19*23^2 + 7*23^3 + 9*23^5+O(23^6)

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,4)(2,3)$ $(1,2,4,3)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $-1$ $1$ $4$ $(1,2,4,3)$ $\zeta_{4}$ $-\zeta_{4}$ $1$ $4$ $(1,3,4,2)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.