# Properties

 Label 3.15864289.42t37.a.a Dimension 3 Group $\GL(3,2)$ Conductor $7^{2} \cdot 569^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $15864289= 7^{2} \cdot 569^{2}$ Artin number field: Splitting field of 7.3.15864289.1 defined by $f= x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ 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 7.3.15864289.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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