# Properties

 Label 1.260.4t1.b.b Dimension $1$ Group $C_4$ Conductor $260$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$260$$$$\medspace = 2^{2} \cdot 5 \cdot 13$$ Artin field: Galois closure of 4.0.4394000.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{260}(83,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 65x^{2} + 260$$ x^4 + 65*x^2 + 260 .

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$7 + 7\cdot 37 + 13\cdot 37^{2} + 4\cdot 37^{3} + 17\cdot 37^{4} +O(37^{5})$$ 7 + 7*37 + 13*37^2 + 4*37^3 + 17*37^4+O(37^5) $r_{ 2 }$ $=$ $$16 + 29\cdot 37 + 35\cdot 37^{2} + 29\cdot 37^{3} + 33\cdot 37^{4} +O(37^{5})$$ 16 + 29*37 + 35*37^2 + 29*37^3 + 33*37^4+O(37^5) $r_{ 3 }$ $=$ $$21 + 7\cdot 37 + 37^{2} + 7\cdot 37^{3} + 3\cdot 37^{4} +O(37^{5})$$ 21 + 7*37 + 37^2 + 7*37^3 + 3*37^4+O(37^5) $r_{ 4 }$ $=$ $$30 + 29\cdot 37 + 23\cdot 37^{2} + 32\cdot 37^{3} + 19\cdot 37^{4} +O(37^{5})$$ 30 + 29*37 + 23*37^2 + 32*37^3 + 19*37^4+O(37^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $1$ $4$ $(1,3,4,2)$ $-\zeta_{4}$ $1$ $4$ $(1,2,4,3)$ $\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.