# Properties

 Label 2.2e2_5_13.4t3.4c1 Dimension 2 Group $D_{4}$ Conductor $2^{2} \cdot 5 \cdot 13$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $260= 2^{2} \cdot 5 \cdot 13$ Artin number field: Splitting field of $f= x^{4} - 3 x^{2} - 2 x + 5$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.2e2_5_13.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 9 + 11\cdot 97 + 23\cdot 97^{2} + 83\cdot 97^{3} + 59\cdot 97^{4} +O\left(97^{ 5 }\right) \\ r_{ 2 } &= 13 + 44\cdot 97 + 62\cdot 97^{2} + 38\cdot 97^{3} + 33\cdot 97^{4} +O\left(97^{ 5 }\right) \\ r_{ 3 } &= 29 + 23\cdot 97 + 65\cdot 97^{2} + 50\cdot 97^{3} + 3\cdot 97^{4} +O\left(97^{ 5 }\right) \\ r_{ 4 } &= 46 + 18\cdot 97 + 43\cdot 97^{2} + 21\cdot 97^{3} +O\left(97^{ 5 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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