# Properties

 Label 3.29768.42t37.a.b Dimension 3 Group $\GL(3,2)$ Conductor $2^{3} \cdot 61^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $29768= 2^{3} \cdot 61^{2}$ Artin number field: Splitting field of 7.3.886133824.1 defined by $f= x^{7} - 2 x^{6} - 4 x^{5} + 6 x^{4} + 8 x^{3} - 22 x^{2} + 16 x - 2$ 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 7.3.886133824.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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