# Properties

 Label 3.8209352.42t37.a.b Dimension 3 Group $\GL(3,2)$ Conductor $2^{3} \cdot 1013^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $8209352= 2^{3} \cdot 1013^{2}$ Artin number field: Splitting field of 7.3.65674816.1 defined by $f= x^{7} - 2 x^{6} + 2 x^{5} - 8 x^{4} + 16 x^{3} - 16 x^{2} + 10 x - 2$ 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 7.3.65674816.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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