# Properties

 Label 2.5_61.4t3.2c1 Dimension 2 Group $D_{4}$ Conductor $5 \cdot 61$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $305= 5 \cdot 61$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} - 5 x + 5$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Even Determinant: 1.5_61.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $2 + 7\cdot 19 + 12\cdot 19^{2} + 17\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 5\cdot 19 + 11\cdot 19^{2} + 14\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $16 + 13\cdot 19 + 7\cdot 19^{2} + 4\cdot 19^{3} + 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $18 + 11\cdot 19 + 6\cdot 19^{2} + 19^{3} + 19^{4} +O\left(19^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.