# Properties

 Label 3.44944.42t37.a.b Dimension 3 Group $\GL(3,2)$ Conductor $2^{4} \cdot 53^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $44944= 2^{4} \cdot 53^{2}$ Artin number field: Splitting field of 7.3.504990784.2 defined by $f= x^{7} - 7 x^{5} - 10 x^{4} + 3 x^{3} + 3 x + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $\PSL(2,7)$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $\PSL(2,7)$ Projective field: Galois closure of 7.3.504990784.2

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{3} + 4 x + 17$
Roots:
 $r_{ 1 }$ $=$ $18 a^{2} + 15 a + 1 + \left(9 a^{2} + 2 a + 5\right)\cdot 19 + \left(5 a^{2} + 5\right)\cdot 19^{2} + \left(4 a^{2} + 10 a + 13\right)\cdot 19^{3} + \left(13 a^{2} + 17 a + 13\right)\cdot 19^{4} + \left(14 a^{2} + 5 a + 1\right)\cdot 19^{5} + \left(10 a^{2} + 10 a + 7\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 2 }$ $=$ $7 + 3\cdot 19 + 16\cdot 19^{2} + 18\cdot 19^{3} + 10\cdot 19^{4} + 10\cdot 19^{5} + 16\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 3 }$ $=$ $5 a^{2} + 8 a + 17 + \left(10 a^{2} + 18 a + 5\right)\cdot 19 + \left(14 a^{2} + a + 10\right)\cdot 19^{2} + \left(11 a^{2} + 18 a + 1\right)\cdot 19^{3} + \left(4 a^{2} + 17 a + 16\right)\cdot 19^{4} + \left(11 a^{2} + 11 a + 17\right)\cdot 19^{5} + \left(15 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 4 }$ $=$ $18 a^{2} + 11 a + 4 + \left(18 a^{2} + 14 a + 1\right)\cdot 19 + \left(5 a^{2} + a + 1\right)\cdot 19^{2} + \left(13 a^{2} + 2 a + 2\right)\cdot 19^{3} + \left(15 a^{2} + 9\right)\cdot 19^{4} + \left(8 a^{2} + 18 a\right)\cdot 19^{5} + \left(12 a^{2} + 3 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 5 }$ $=$ $12 a^{2} + a + 7 + \left(a^{2} + 14 a + 18\right)\cdot 19 + \left(3 a^{2} + 11 a + 5\right)\cdot 19^{2} + \left(8 a^{2} + 11 a + 7\right)\cdot 19^{3} + \left(3 a^{2} + 10 a + 14\right)\cdot 19^{4} + \left(5 a^{2} + 16 a + 9\right)\cdot 19^{5} + \left(16 a^{2} + 10 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 6 }$ $=$ $8 a^{2} + 7 a + 9 + \left(17 a^{2} + 9 a + 3\right)\cdot 19 + \left(9 a^{2} + 5 a + 5\right)\cdot 19^{2} + \left(16 a^{2} + 5 a + 4\right)\cdot 19^{3} + \left(18 a^{2} + 8 a + 11\right)\cdot 19^{4} + \left(4 a^{2} + 3 a + 15\right)\cdot 19^{5} + \left(9 a^{2} + 4 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 7 }$ $=$ $15 a^{2} + 15 a + 12 + \left(17 a^{2} + 16 a\right)\cdot 19 + \left(17 a^{2} + 16 a + 13\right)\cdot 19^{2} + \left(2 a^{2} + 9 a + 9\right)\cdot 19^{3} + \left(a^{2} + 2 a\right)\cdot 19^{4} + \left(12 a^{2} + a + 1\right)\cdot 19^{5} + \left(7 a^{2} + 12 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $3$ $21$ $2$ $(3,5)(4,7)$ $-1$ $56$ $3$ $(1,4,5)(2,6,7)$ $0$ $42$ $4$ $(1,7,5,6)(2,3)$ $1$ $24$ $7$ $(1,7,4,5,2,3,6)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $24$ $7$ $(1,5,6,4,3,7,2)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.