Properties

 Label 2.68.4t3.b.a Dimension 2 Group $D_{4}$ Conductor $2^{2} \cdot 17$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $68= 2^{2} \cdot 17$ Artin number field: Splitting field of 4.2.1156.1 defined by $f= x^{4} - x^{3} - 2 x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.68.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $12 + 53 + 17\cdot 53^{3} + 24\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $28 + 39\cdot 53 + 53^{2} + 5\cdot 53^{3} + 11\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $31 + 41\cdot 53 + 9\cdot 53^{2} + 39\cdot 53^{3} + 12\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $36 + 23\cdot 53 + 41\cdot 53^{2} + 44\cdot 53^{3} + 4\cdot 53^{4} +O\left(53^{ 5 }\right)$

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

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

Character values on conjugacy classes

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