# Properties

 Label 3.2e3_1733e2.42t37.1c2 Dimension 3 Group $\GL(3,2)$ Conductor $2^{3} \cdot 1733^{2}$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $24026312= 2^{3} \cdot 1733^{2}$ Artin number field: Splitting field of $f= x^{7} - 4 x^{5} - 2 x^{4} - 4 x^{3} - 8 x^{2} - 4 x - 4$ over $\Q$ Size of Galois orbit: 2 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 $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{3} + x + 40$
Roots:
 $r_{ 1 }$ $=$ $30 a + 40 + \left(13 a^{2} + 3 a + 4\right)\cdot 43 + \left(30 a^{2} + 22 a + 21\right)\cdot 43^{2} + \left(21 a^{2} + 21 a\right)\cdot 43^{3} + \left(30 a^{2} + 18\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $38 a^{2} + 5 a + 15 + \left(16 a^{2} + 34 a + 3\right)\cdot 43 + \left(26 a^{2} + 23 a + 2\right)\cdot 43^{2} + \left(14 a^{2} + 11 a + 32\right)\cdot 43^{3} + \left(13 a^{2} + 15 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $40 + 34\cdot 43 + 18\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $8 a^{2} + 21 a + 31 + \left(24 a^{2} + 20 a + 26\right)\cdot 43 + \left(31 a + 15\right)\cdot 43^{2} + \left(33 a^{2} + 13 a + 22\right)\cdot 43^{3} + \left(24 a^{2} + 4 a + 28\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 5 }$ $=$ $17 a^{2} + 15 a + 1 + \left(12 a^{2} + 14 a + 29\right)\cdot 43 + \left(26 a^{2} + 3 a + 30\right)\cdot 43^{2} + \left(27 a^{2} + 14 a + 40\right)\cdot 43^{3} + \left(31 a^{2} + 40 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 6 }$ $=$ $35 a^{2} + 35 a + 6 + \left(5 a^{2} + 18 a\right)\cdot 43 + \left(12 a^{2} + 32 a + 9\right)\cdot 43^{2} + \left(31 a^{2} + 7 a + 21\right)\cdot 43^{3} + \left(30 a^{2} + 38 a + 32\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 7 }$ $=$ $31 a^{2} + 23 a + 39 + \left(13 a^{2} + 37 a + 29\right)\cdot 43 + \left(33 a^{2} + 15 a + 6\right)\cdot 43^{2} + \left(17 a + 37\right)\cdot 43^{3} + \left(41 a^{2} + 30 a + 9\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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