Properties

 Label 2.2e8_3.4t3.6c1 Dimension 2 Group $D_4$ Conductor $2^{8} \cdot 3$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $D_4$ Conductor: $768= 2^{8} \cdot 3$ Artin number field: Splitting field of $f= x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 2 + 4\cdot 67 + 27\cdot 67^{2} + 18\cdot 67^{3} + 3\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 2 } &= 9 + 64\cdot 67 + 67^{2} + 47\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 3 } &= 12 + 31\cdot 67 + 26\cdot 67^{2} + 12\cdot 67^{3} + 24\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 4 } &= 13 + 19\cdot 67 + 3\cdot 67^{2} + 25\cdot 67^{3} + 4\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 5 } &= 54 + 47\cdot 67 + 63\cdot 67^{2} + 41\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 6 } &= 55 + 35\cdot 67 + 40\cdot 67^{2} + 54\cdot 67^{3} + 42\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 7 } &= 58 + 2\cdot 67 + 65\cdot 67^{2} + 19\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right) \\ r_{ 8 } &= 65 + 62\cdot 67 + 39\cdot 67^{2} + 48\cdot 67^{3} + 63\cdot 67^{4} +O\left(67^{ 5 }\right) \\ \end{aligned}

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

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

Character values on conjugacy classes

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