# Properties

 Label 8.7e4_569e4.21t14.2c1 Dimension 8 Group $\GL(3,2)$ Conductor $7^{4} \cdot 569^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $8$ Group: $\GL(3,2)$ Conductor: $251675665475521= 7^{4} \cdot 569^{4}$ Artin number field: Splitting field of $f= x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1$ 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 $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{3} + 2 x + 11$
Roots:
 $r_{ 1 }$ $=$ $11 a^{2} + 7 a + 10 + \left(a^{2} + 12 a + 4\right)\cdot 13 + \left(4 a^{2} + 10 a + 12\right)\cdot 13^{2} + \left(2 a^{2} + a + 9\right)\cdot 13^{3} + \left(a + 1\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 a^{2} + 12 a + 7 + \left(8 a^{2} + 7 a + 5\right)\cdot 13 + \left(12 a^{2} + 3 a + 6\right)\cdot 13^{2} + \left(6 a^{2} + 3 a + 7\right)\cdot 13^{3} + \left(9 a^{2} + 12 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $3 a^{2} + 7 a + 8 + \left(2 a^{2} + 5 a + 9\right)\cdot 13 + \left(9 a^{2} + 11 a + 1\right)\cdot 13^{2} + \left(3 a^{2} + 7 a + 3\right)\cdot 13^{3} + \left(3 a^{2} + 12 a + 10\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $8 a^{2} + a + 1 + \left(5 a + 4\right)\cdot 13 + \left(8 a^{2} + 7 a + 5\right)\cdot 13^{2} + \left(4 a^{2} + a + 11\right)\cdot 13^{3} + \left(8 a^{2} + 8 a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 5 }$ $=$ $12 a^{2} + 2 + \left(5 a^{2} + 2 a + 11\right)\cdot 13 + \left(4 a^{2} + 6 a + 4\right)\cdot 13^{2} + \left(2 a^{2} + 7 a + 8\right)\cdot 13^{3} + \left(5 a^{2} + 5 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 6 }$ $=$ $5 + 9\cdot 13 + 8\cdot 13^{2} + 2\cdot 13^{3} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 7 }$ $=$ $6 a^{2} + 12 a + 7 + \left(6 a^{2} + 5 a + 7\right)\cdot 13 + \left(12 a + 12\right)\cdot 13^{2} + \left(6 a^{2} + 3 a + 8\right)\cdot 13^{3} + \left(12 a^{2} + 12 a + 6\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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