Properties

 Label 2.55.4t3.c.a Dimension 2 Group $D_{4}$ Conductor $5 \cdot 11$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $55= 5 \cdot 11$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.55.2t1.a.a

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 27\cdot 59 + 50\cdot 59^{2} + 50\cdot 59^{3} + 9\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 + 4\cdot 59 + 33\cdot 59^{2} + 33\cdot 59^{3} + 2\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $18 + 49\cdot 59 + 51\cdot 59^{2} + 11\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 4 }$ $=$ $31 + 37\cdot 59 + 41\cdot 59^{2} + 21\cdot 59^{3} + 12\cdot 59^{4} +O\left(59^{ 5 }\right)$

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.