Properties

 Label 2.5_11_29.4t3.4c1 Dimension 2 Group $D_{4}$ Conductor $5 \cdot 11 \cdot 29$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $1595= 5 \cdot 11 \cdot 29$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 92$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.5_11_29.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $24 + 3\cdot 61 + 43\cdot 61^{2} + 16\cdot 61^{3} + 8\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 2 }$ $=$ $25 + 54\cdot 61 + 50\cdot 61^{2} + 36\cdot 61^{3} +O\left(61^{ 5 }\right)$ $r_{ 3 }$ $=$ $37 + 6\cdot 61 + 10\cdot 61^{2} + 24\cdot 61^{3} + 60\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 4 }$ $=$ $38 + 57\cdot 61 + 17\cdot 61^{2} + 44\cdot 61^{3} + 52\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

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.