# Properties

 Label 3.5e2_653e2.42t37.1 Dimension 3 Group $\GL(3,2)$ Conductor $5^{2} \cdot 653^{2}$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $10660225= 5^{2} \cdot 653^{2}$ Artin number field: Splitting field of $f= x^{7} - 3 x^{6} + 3 x^{5} + 6 x^{4} - 17 x^{3} + 6 x^{2} + 16 x - 8$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $\PSL(2,7)$ Parity: Even

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $21$ $2$ $(1,5)(6,7)$ $-1$ $-1$ $56$ $3$ $(1,3,5)(4,7,6)$ $0$ $0$ $42$ $4$ $(1,2)(3,4,5,7)$ $1$ $1$ $24$ $7$ $(1,7,6,3,4,5,2)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $24$ $7$ $(1,3,2,6,5,7,4)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.