Properties

 Label 2.2e4_5_19.4t3.7c1 Dimension 2 Group $D_4$ Conductor $2^{4} \cdot 5 \cdot 19$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $D_4$ Conductor: $1520= 2^{4} \cdot 5 \cdot 19$ Artin number field: Splitting field of $f= x^{8} - 2 x^{7} - 38 x^{6} + 2 x^{5} + 458 x^{4} + 562 x^{3} - 1438 x^{2} - 3402 x - 1899$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Even Determinant: 1.2e2_5_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 12\cdot 61 + 52\cdot 61^{2} + 52\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 2 }$ $=$ $5 + 57\cdot 61 + 35\cdot 61^{2} + 57\cdot 61^{3} + 42\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 + 49\cdot 61 + 31\cdot 61^{2} + 19\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 4 }$ $=$ $29 + 21\cdot 61 + 9\cdot 61^{2} + 24\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 5 }$ $=$ $42 + 47\cdot 61 + 2\cdot 61^{2} + 51\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 6 }$ $=$ $45 + 36\cdot 61 + 19\cdot 61^{2} + 44\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 7 }$ $=$ $53 + 2\cdot 61 + 25\cdot 61^{2} + 18\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 8 }$ $=$ $55 + 16\cdot 61 + 6\cdot 61^{2} + 37\cdot 61^{3} + 11\cdot 61^{4} +O\left(61^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,2)(3,7)(4,6)(5,8)$ $(1,3)(2,6)(4,5)(7,8)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,5)(2,8)(3,4)(6,7)$ $-2$ $2$ $2$ $(1,2)(3,7)(4,6)(5,8)$ $0$ $2$ $2$ $(1,3)(2,6)(4,5)(7,8)$ $0$ $2$ $4$ $(1,6,5,7)(2,3,8,4)$ $0$
The blue line marks the conjugacy class containing complex conjugation.