# Properties

 Label 1.260.4t1.a.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 4.4.878800.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{260}(239,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 65x^{2} + 325$$ x^4 - 65*x^2 + 325 .

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$4 + 14\cdot 17 + 17^{2} + 2\cdot 17^{4} + 8\cdot 17^{5} +O(17^{6})$$ 4 + 14*17 + 17^2 + 2*17^4 + 8*17^5+O(17^6) $r_{ 2 }$ $=$ $$7 + 9\cdot 17 + 9\cdot 17^{2} + 15\cdot 17^{3} + 12\cdot 17^{4} + 9\cdot 17^{5} +O(17^{6})$$ 7 + 9*17 + 9*17^2 + 15*17^3 + 12*17^4 + 9*17^5+O(17^6) $r_{ 3 }$ $=$ $$10 + 7\cdot 17 + 7\cdot 17^{2} + 17^{3} + 4\cdot 17^{4} + 7\cdot 17^{5} +O(17^{6})$$ 10 + 7*17 + 7*17^2 + 17^3 + 4*17^4 + 7*17^5+O(17^6) $r_{ 4 }$ $=$ $$13 + 2\cdot 17 + 15\cdot 17^{2} + 16\cdot 17^{3} + 14\cdot 17^{4} + 8\cdot 17^{5} +O(17^{6})$$ 13 + 2*17 + 15*17^2 + 16*17^3 + 14*17^4 + 8*17^5+O(17^6)

## 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 value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $1$ $4$ $(1,2,4,3)$ $-\zeta_{4}$ $1$ $4$ $(1,3,4,2)$ $\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.