# Properties

 Label 1.1020.4t1.c.a Dimension 1 Group $C_4$ Conductor $2^{2} \cdot 3 \cdot 5 \cdot 17$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $1020= 2^{2} \cdot 3 \cdot 5 \cdot 17$ Artin number field: Splitting field of $f= x^{4} + 255 x^{2} + 13005$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{1020}(407,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 7.
Roots:
 $r_{ 1 }$ $=$ $2 + 3\cdot 19 + 11\cdot 19^{2} + 16\cdot 19^{3} + 16\cdot 19^{4} + 3\cdot 19^{5} + 2\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 2 }$ $=$ $8 + 16\cdot 19 + 13\cdot 19^{2} + 14\cdot 19^{3} + 3\cdot 19^{4} + 19^{5} + 14\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 3 }$ $=$ $11 + 2\cdot 19 + 5\cdot 19^{2} + 4\cdot 19^{3} + 15\cdot 19^{4} + 17\cdot 19^{5} + 4\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 4 }$ $=$ $17 + 15\cdot 19 + 7\cdot 19^{2} + 2\cdot 19^{3} + 2\cdot 19^{4} + 15\cdot 19^{5} + 16\cdot 19^{6} +O\left(19^{ 7 }\right)$

### 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.