# Properties

 Label 8.2e10_743e4.21t14.1c1 Dimension 8 Group $\GL(3,2)$ Conductor $2^{10} \cdot 743^{4}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $8$ Group: $\GL(3,2)$ Conductor: $312072292762624= 2^{10} \cdot 743^{4}$ Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PSL(2,7)$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{3} + 2 x + 27$
Roots:
 $r_{ 1 }$ $=$ $6 a^{2} + 5 a + 13 + \left(19 a^{2} + 17 a + 3\right)\cdot 29 + \left(4 a^{2} + 23\right)\cdot 29^{2} + \left(20 a^{2} + 25 a + 11\right)\cdot 29^{3} + \left(5 a^{2} + 5 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 a^{2} + 13 a + 6 + \left(16 a^{2} + 28 a + 9\right)\cdot 29 + \left(25 a^{2} + 17 a + 12\right)\cdot 29^{2} + \left(11 a^{2} + 18 a + 10\right)\cdot 29^{3} + \left(4 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 a^{2} + 9 a + 25 + \left(15 a^{2} + 13 a + 27\right)\cdot 29 + \left(5 a^{2} + 18 a + 14\right)\cdot 29^{2} + \left(6 a^{2} + 26 a + 10\right)\cdot 29^{3} + \left(4 a^{2} + 11 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $15 a^{2} + 11 a + 25 + \left(22 a^{2} + 12 a + 7\right)\cdot 29 + \left(27 a^{2} + 10 a + 15\right)\cdot 29^{2} + \left(25 a^{2} + 14 a + 19\right)\cdot 29^{3} + \left(22 a^{2} + 18 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 5 }$ $=$ $17 + 15\cdot 29 + 13\cdot 29^{2} + 9\cdot 29^{3} + 2\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 6 }$ $=$ $20 a^{2} + 22 a + 26 + \left(17 a^{2} + 21 a + 1\right)\cdot 29 + \left(25 a^{2} + 20 a + 3\right)\cdot 29^{2} + \left(11 a^{2} + a + 18\right)\cdot 29^{3} + \left(7 a^{2} + 23 a\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 7 }$ $=$ $26 a^{2} + 27 a + 5 + \left(24 a^{2} + 22 a + 21\right)\cdot 29 + \left(26 a^{2} + 18 a + 4\right)\cdot 29^{2} + \left(10 a^{2} + 7\right)\cdot 29^{3} + \left(17 a^{2} + 23 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $8$ $21$ $2$ $(1,7)(2,6)$ $0$ $56$ $3$ $(1,3,5)(4,7,6)$ $-1$ $42$ $4$ $(1,6,5,4)(2,3)$ $0$ $24$ $7$ $(1,7,6,3,2,5,4)$ $1$ $24$ $7$ $(1,3,4,6,5,7,2)$ $1$
The blue line marks the conjugacy class containing complex conjugation.