# Properties

 Label 1.40.4t1.b.b Dimension 1 Group $C_4$ Conductor $2^{3} \cdot 5$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $40= 2^{3} \cdot 5$ Artin number field: Splitting field of 4.0.8000.2 defined by $f= x^{4} + 10 x^{2} + 20$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{40}(37,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $2 + 19 + 12\cdot 19^{3} + 2\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 + 9\cdot 19 + 15\cdot 19^{2} + 14\cdot 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 + 9\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $17 + 17\cdot 19 + 18\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} +O\left(19^{ 5 }\right)$

### 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.