# Properties

 Label 3.1569992.42t37.a.a Dimension 3 Group $\GL(3,2)$ Conductor $2^{3} \cdot 443^{2}$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $1569992= 2^{3} \cdot 443^{2}$ Artin number field: Splitting field of 7.3.12559936.2 defined by $f= x^{7} - 3 x^{6} + x^{5} + 3 x^{4} - x^{3} + x^{2} - 3 x - 1$ 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 7.3.12559936.2

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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