# Properties

 Label 1.240.4t1.d.b Dimension $1$ Group $C_4$ Conductor $240$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$240$$$$\medspace = 2^{4} \cdot 3 \cdot 5$$ Artin field: 4.0.2304000.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{240}(83,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 60 x^{2} + 90$$  .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$14 + 6\cdot 41 + 10\cdot 41^{2} + 33\cdot 41^{3} + 18\cdot 41^{4} +O(41^{5})$$ $r_{ 2 }$ $=$ $$20 + 20\cdot 41 + 2\cdot 41^{2} + 37\cdot 41^{3} + 6\cdot 41^{4} +O(41^{5})$$ $r_{ 3 }$ $=$ $$21 + 20\cdot 41 + 38\cdot 41^{2} + 3\cdot 41^{3} + 34\cdot 41^{4} +O(41^{5})$$ $r_{ 4 }$ $=$ $$27 + 34\cdot 41 + 30\cdot 41^{2} + 7\cdot 41^{3} + 22\cdot 41^{4} +O(41^{5})$$

## 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.