# Properties

 Label 1.260.4t1.d.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 4.4.338000.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{260}(103,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 65x^{2} + 845$$ x^4 - 65*x^2 + 845 .

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$1 + 11 + 7\cdot 11^{2} + 11^{3} + 8\cdot 11^{4} + 2\cdot 11^{5} + 6\cdot 11^{6} +O(11^{7})$$ 1 + 11 + 7*11^2 + 11^3 + 8*11^4 + 2*11^5 + 6*11^6+O(11^7) $r_{ 2 }$ $=$ $$3 + 6\cdot 11 + 2\cdot 11^{2} + 7\cdot 11^{3} + 4\cdot 11^{4} + 10\cdot 11^{5} + 9\cdot 11^{6} +O(11^{7})$$ 3 + 6*11 + 2*11^2 + 7*11^3 + 4*11^4 + 10*11^5 + 9*11^6+O(11^7) $r_{ 3 }$ $=$ $$8 + 4\cdot 11 + 8\cdot 11^{2} + 3\cdot 11^{3} + 6\cdot 11^{4} + 11^{6} +O(11^{7})$$ 8 + 4*11 + 8*11^2 + 3*11^3 + 6*11^4 + 11^6+O(11^7) $r_{ 4 }$ $=$ $$10 + 9\cdot 11 + 3\cdot 11^{2} + 9\cdot 11^{3} + 2\cdot 11^{4} + 8\cdot 11^{5} + 4\cdot 11^{6} +O(11^{7})$$ 10 + 9*11 + 3*11^2 + 9*11^3 + 2*11^4 + 8*11^5 + 4*11^6+O(11^7)

## 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.