Properties

 Label 2.1224.24t22.a.b Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $2^{3} \cdot 3^{2} \cdot 17$ Root number not computed Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $1224= 2^{3} \cdot 3^{2} \cdot 17$ Artin number field: Splitting field of 8.2.11002604544.1 defined by $f= x^{8} - 6 x^{4} - 24 x^{2} - 51$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.51.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.7344.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $x^{2} + 58 x + 2$
Roots:
 $r_{ 1 }$ $=$ $40 + 49\cdot 59 + 13\cdot 59^{2} + 5\cdot 59^{3} + 28\cdot 59^{4} + 54\cdot 59^{5} + 16\cdot 59^{6} + 26\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 2 }$ $=$ $8 a + 55 + \left(52 a + 36\right)\cdot 59 + \left(34 a + 8\right)\cdot 59^{2} + \left(4 a + 15\right)\cdot 59^{3} + \left(46 a + 38\right)\cdot 59^{4} + \left(11 a + 46\right)\cdot 59^{5} + \left(30 a + 49\right)\cdot 59^{6} + \left(45 a + 21\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 3 }$ $=$ $8 a + 24 + \left(8 a + 33\right)\cdot 59 + \left(28 a + 17\right)\cdot 59^{2} + \left(40 a + 44\right)\cdot 59^{3} + \left(38 a + 26\right)\cdot 59^{4} + \left(29 a + 48\right)\cdot 59^{5} + \left(a + 53\right)\cdot 59^{6} + \left(7 a + 42\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 4 }$ $=$ $8 a + 27 + \left(8 a + 25\right)\cdot 59 + \left(28 a + 21\right)\cdot 59^{2} + \left(40 a + 2\right)\cdot 59^{3} + \left(38 a + 34\right)\cdot 59^{4} + \left(29 a + 19\right)\cdot 59^{5} + \left(a + 33\right)\cdot 59^{6} + \left(7 a + 10\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 5 }$ $=$ $19 + 9\cdot 59 + 45\cdot 59^{2} + 53\cdot 59^{3} + 30\cdot 59^{4} + 4\cdot 59^{5} + 42\cdot 59^{6} + 32\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 6 }$ $=$ $51 a + 4 + \left(6 a + 22\right)\cdot 59 + \left(24 a + 50\right)\cdot 59^{2} + \left(54 a + 43\right)\cdot 59^{3} + \left(12 a + 20\right)\cdot 59^{4} + \left(47 a + 12\right)\cdot 59^{5} + \left(28 a + 9\right)\cdot 59^{6} + \left(13 a + 37\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 7 }$ $=$ $51 a + 35 + \left(50 a + 25\right)\cdot 59 + \left(30 a + 41\right)\cdot 59^{2} + \left(18 a + 14\right)\cdot 59^{3} + \left(20 a + 32\right)\cdot 59^{4} + \left(29 a + 10\right)\cdot 59^{5} + \left(57 a + 5\right)\cdot 59^{6} + \left(51 a + 16\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ $r_{ 8 }$ $=$ $51 a + 32 + \left(50 a + 33\right)\cdot 59 + \left(30 a + 37\right)\cdot 59^{2} + \left(18 a + 56\right)\cdot 59^{3} + \left(20 a + 24\right)\cdot 59^{4} + \left(29 a + 39\right)\cdot 59^{5} + \left(57 a + 25\right)\cdot 59^{6} + \left(51 a + 48\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$

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

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

Character values on conjugacy classes

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