# Properties

 Label 1.40.4t1.b.b Dimension $1$ Group $C_4$ Conductor $40$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$40$$$$\medspace = 2^{3} \cdot 5$$ Artin field: Galois closure of 4.0.8000.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{40}(37,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 10x^{2} + 20$$ x^4 + 10*x^2 + 20 .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$2 + 19 + 12\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})$$ 2 + 19 + 12*19^3 + 2*19^4+O(19^5) $r_{ 2 }$ $=$ $$9 + 9\cdot 19 + 15\cdot 19^{2} + 14\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})$$ 9 + 9*19 + 15*19^2 + 14*19^3 + 8*19^4+O(19^5) $r_{ 3 }$ $=$ $$10 + 9\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})$$ 10 + 9*19 + 3*19^2 + 4*19^3 + 10*19^4+O(19^5) $r_{ 4 }$ $=$ $$17 + 17\cdot 19 + 18\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})$$ 17 + 17*19 + 18*19^2 + 6*19^3 + 16*19^4+O(19^5)

## 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.