# Properties

 Label 3.12759184.42t37.a Dimension $3$ Group $\GL(3,2)$ Conductor $12759184$ Indicator $0$

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## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $$12759184$$$$\medspace = 2^{4} \cdot 19^{2} \cdot 47^{2}$$ Artin number field: Galois closure of 7.7.18424261696.1 Galois orbit size: $2$ Smallest permutation container: $\PSL(2,7)$ Parity: even Projective image: $\PSL(2,7)$ Projective field: Galois closure of 7.7.18424261696.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{3} + 6 x + 35$
Roots:
 $r_{ 1 }$ $=$ $3 a^{2} + 27 a + 14 + \left(34 a^{2} + 3 a + 28\right)\cdot 37 + \left(35 a^{2} + 2 a + 17\right)\cdot 37^{2} + \left(19 a^{2} + 12 a + 34\right)\cdot 37^{3} + \left(11 a^{2} + 32 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 a^{2} + 21 + \left(25 a^{2} + 15 a + 16\right)\cdot 37 + \left(17 a^{2} + 11 a + 4\right)\cdot 37^{2} + \left(7 a^{2} + 15 a + 22\right)\cdot 37^{3} + \left(3 a^{2} + 32 a + 23\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $33 a^{2} + 31 a + 35 + \left(34 a^{2} + 29 a + 17\right)\cdot 37 + \left(21 a^{2} + 17 a + 21\right)\cdot 37^{2} + \left(33 a^{2} + 23 a + 15\right)\cdot 37^{3} + \left(20 a^{2} + 28 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $26 + 22\cdot 37 + 21\cdot 37^{2} + 11\cdot 37^{3} + 20\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $5 a^{2} + 35 a + 22 + \left(6 a^{2} + 26 a + 27\right)\cdot 37 + \left(20 a^{2} + 4 a + 28\right)\cdot 37^{2} + \left(18 a^{2} + 35 a + 28\right)\cdot 37^{3} + \left(21 a^{2} + 27 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 6 }$ $=$ $29 a^{2} + 12 a + 7 + \left(33 a^{2} + 6 a + 27\right)\cdot 37 + \left(17 a^{2} + 30 a + 19\right)\cdot 37^{2} + \left(35 a^{2} + 26 a + 22\right)\cdot 37^{3} + \left(3 a^{2} + 13 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 7 }$ $=$ $30 a^{2} + 6 a + 23 + \left(13 a^{2} + 29 a + 7\right)\cdot 37 + \left(34 a^{2} + 7 a + 34\right)\cdot 37^{2} + \left(32 a^{2} + 35 a + 12\right)\cdot 37^{3} + \left(12 a^{2} + 12 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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