# Properties

 Label 6.18424261696.7t5.a.a Dimension $6$ Group $\GL(3,2)$ Conductor $18424261696$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $\GL(3,2)$ Conductor: $$18424261696$$$$\medspace = 2^{6} \cdot 19^{4} \cdot 47^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 7.7.18424261696.1 Galois orbit size: $1$ Smallest permutation container: $\GL(3,2)$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $\PSL(2,7)$ Projective field: Galois closure of 7.7.18424261696.1

## Defining polynomial

 $f(x)$ $=$ $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$.

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 value $1$ $1$ $()$ $6$ $21$ $2$ $(1,4)(5,6)$ $2$ $56$ $3$ $(1,2,4)(3,5,6)$ $0$ $42$ $4$ $(1,6,4,5)(2,7)$ $0$ $24$ $7$ $(1,7,2,6,3,4,5)$ $-1$ $24$ $7$ $(1,6,5,2,4,7,3)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.