# Properties

 Label 2.2e3_3e2_5e2.24t22.1c2 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{3} \cdot 3^{2} \cdot 5^{2}$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $1800= 2^{3} \cdot 3^{2} \cdot 5^{2}$ Artin number field: Splitting field of $f= x^{8} - 6 x^{4} - 12 x^{2} - 3$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.3.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$
Roots:
 $r_{ 1 }$ $=$ $15 a + 36 + \left(6 a + 6\right)\cdot 41 + \left(7 a + 12\right)\cdot 41^{2} + \left(15 a + 26\right)\cdot 41^{3} + \left(37 a + 26\right)\cdot 41^{4} + \left(40 a + 12\right)\cdot 41^{5} + \left(10 a + 1\right)\cdot 41^{6} + \left(39 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 2 }$ $=$ $15 a + 1 + \left(6 a + 30\right)\cdot 41 + \left(7 a + 13\right)\cdot 41^{2} + \left(15 a + 17\right)\cdot 41^{3} + \left(37 a + 40\right)\cdot 41^{4} + \left(40 a + 24\right)\cdot 41^{5} + \left(10 a + 6\right)\cdot 41^{6} + \left(39 a + 40\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 3 }$ $=$ $24 a + 5 + \left(14 a + 31\right)\cdot 41 + \left(36 a + 34\right)\cdot 41^{2} + \left(35 a + 25\right)\cdot 41^{3} + \left(7 a + 26\right)\cdot 41^{4} + \left(28 a + 2\right)\cdot 41^{5} + \left(25 a + 37\right)\cdot 41^{6} + \left(2 a + 8\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 4 }$ $=$ $18 + 7\cdot 41 + 31\cdot 41^{2} + 19\cdot 41^{3} + 25\cdot 41^{4} + 18\cdot 41^{5} + 7\cdot 41^{6} + 12\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 5 }$ $=$ $26 a + 5 + \left(34 a + 34\right)\cdot 41 + \left(33 a + 28\right)\cdot 41^{2} + \left(25 a + 14\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} + 28\cdot 41^{5} + \left(30 a + 39\right)\cdot 41^{6} + \left(a + 23\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 6 }$ $=$ $26 a + 40 + \left(34 a + 10\right)\cdot 41 + \left(33 a + 27\right)\cdot 41^{2} + \left(25 a + 23\right)\cdot 41^{3} + 3 a\cdot 41^{4} + 16\cdot 41^{5} + \left(30 a + 34\right)\cdot 41^{6} + a\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 7 }$ $=$ $17 a + 36 + \left(26 a + 9\right)\cdot 41 + \left(4 a + 6\right)\cdot 41^{2} + \left(5 a + 15\right)\cdot 41^{3} + \left(33 a + 14\right)\cdot 41^{4} + \left(12 a + 38\right)\cdot 41^{5} + \left(15 a + 3\right)\cdot 41^{6} + \left(38 a + 32\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ $r_{ 8 }$ $=$ $23 + 33\cdot 41 + 9\cdot 41^{2} + 21\cdot 41^{3} + 15\cdot 41^{4} + 22\cdot 41^{5} + 33\cdot 41^{6} + 28\cdot 41^{7} +O\left(41^{ 8 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$ $-2$ $12$ $2$ $(1,5)(2,8)(4,6)$ $0$ $8$ $3$ $(1,4,7)(3,5,8)$ $-1$ $6$ $4$ $(1,3,5,7)(2,8,6,4)$ $0$ $8$ $6$ $(1,3,4,5,7,8)(2,6)$ $1$ $6$ $8$ $(1,4,3,2,5,8,7,6)$ $\zeta_{8}^{3} + \zeta_{8}$ $6$ $8$ $(1,8,3,6,5,4,7,2)$ $-\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.