# Properties

 Label 1.65.4t1.d.a Dimension $1$ Group $C_4$ Conductor $65$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$65$$$$\medspace = 5 \cdot 13$$ Artin number field: Galois closure of 4.4.274625.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{65}(8,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{4} - x^{3} - 24 x^{2} + 4 x + 16$.

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $12 + 42\cdot 61 + 60\cdot 61^{2} + 58\cdot 61^{3} + 30\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 + 11\cdot 61 + 33\cdot 61^{2} + 35\cdot 61^{3} + 11\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 3 }$ $=$ $45 + 61 + 11\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 4 }$ $=$ $46 + 5\cdot 61 + 17\cdot 61^{2} + 54\cdot 61^{3} + 40\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.