# Properties

 Label 3.7513081.42t37.a.a Dimension 3 Group $\GL(3,2)$ Conductor $2741^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $7513081= 2741^{2}$ Artin number field: Splitting field of 7.3.7513081.1 defined by $f= x^{7} - x^{6} - 4 x^{5} + x^{4} + 4 x^{3} - 3 x + 1$ 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 7.3.7513081.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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