# Properties

 Label 1.29.4t1.a Dimension $1$ Group $C_4$ Conductor $29$ Indicator $0$

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## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$29$$ Artin number field: Galois closure of 4.0.24389.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd 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 5.
Roots:
 $r_{ 1 }$ $=$ $$15\cdot 23 + 11\cdot 23^{2} + 16\cdot 23^{3} + 5\cdot 23^{4} +O(23^{5})$$ 15*23 + 11*23^2 + 16*23^3 + 5*23^4+O(23^5) $r_{ 2 }$ $=$ $$6 + 10\cdot 23 + 21\cdot 23^{2} + 4\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})$$ 6 + 10*23 + 21*23^2 + 4*23^3 + 19*23^4+O(23^5) $r_{ 3 }$ $=$ $$8 + 11\cdot 23^{2} + 9\cdot 23^{3} +O(23^{5})$$ 8 + 11*23^2 + 9*23^3+O(23^5) $r_{ 4 }$ $=$ $$10 + 20\cdot 23 + 23^{2} + 15\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})$$ 10 + 20*23 + 23^2 + 15*23^3 + 20*23^4+O(23^5)

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

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

### Character values on conjugacy classes

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