# Properties

 Label 2.3e2_7e2_11e2.8t5.2c1 Dimension 2 Group $Q_8$ Conductor $3^{2} \cdot 7^{2} \cdot 11^{2}$ Root number 1 Frobenius-Schur indicator -1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $53361= 3^{2} \cdot 7^{2} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{8} - x^{7} + 50 x^{6} + 71 x^{5} + 529 x^{4} + 2173 x^{3} + 842 x^{2} + 5545 x + 17623$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $6 + 22\cdot 67 + 21\cdot 67^{2} + 22\cdot 67^{3} + 56\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $14 + 4\cdot 67 + 3\cdot 67^{2} + 14\cdot 67^{3} + 4\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $17 + 49\cdot 67 + 36\cdot 67^{2} + 12\cdot 67^{3} + 38\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $18 + 14\cdot 67 + 51\cdot 67^{2} + 19\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 5 }$ $=$ $21 + 30\cdot 67 + 51\cdot 67^{2} + 62\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 6 }$ $=$ $34 + 25\cdot 67 + 41\cdot 67^{2} + 51\cdot 67^{3} + 22\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 7 }$ $=$ $38 + 33\cdot 67 + 7\cdot 67^{2} + 41\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 8 }$ $=$ $54 + 21\cdot 67 + 55\cdot 67^{2} + 43\cdot 67^{3} + 18\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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