# Properties

 Label 3.2e3_1237e2.42t37.1c1 Dimension 3 Group $\GL(3,2)$ Conductor $2^{3} \cdot 1237^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $12241352= 2^{3} \cdot 1237^{2}$ Artin number field: Splitting field of $f= x^{7} + 2 x^{5} - 12 x^{4} + 10 x^{3} - 14 x^{2} + 10 x - 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $\PSL(2,7)$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{3} + x + 14$
Roots:
 $r_{ 1 }$ $=$ $5 a^{2} + 4 a + 1 + \left(9 a^{2} + 15 a\right)\cdot 17 + \left(2 a^{2} + 6 a + 16\right)\cdot 17^{2} + \left(6 a^{2} + 14 a + 4\right)\cdot 17^{3} + \left(11 a^{2} + 12 a + 11\right)\cdot 17^{4} + \left(7 a^{2} + a + 13\right)\cdot 17^{5} + \left(10 a^{2} + 15 a + 5\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 2 }$ $=$ $12 a^{2} + 10 a + \left(10 a^{2} + a + 1\right)\cdot 17 + \left(2 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(8 a^{2} + 7 a + 11\right)\cdot 17^{3} + \left(3 a^{2} + 11\right)\cdot 17^{4} + \left(16 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(5 a^{2} + 3 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 3 }$ $=$ $16 + 15\cdot 17 + 12\cdot 17^{2} + 5\cdot 17^{3} + 16\cdot 17^{4} + 9\cdot 17^{5} + 14\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 4 }$ $=$ $a + 14 + \left(a^{2} + 14 a + 12\right)\cdot 17 + \left(7 a^{2} + 5 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + a + 11\right)\cdot 17^{3} + \left(2 a^{2} + 4 a + 3\right)\cdot 17^{4} + \left(7 a^{2} + 16 a + 4\right)\cdot 17^{5} + \left(10 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 5 }$ $=$ $13 a^{2} + 7 a + \left(4 a^{2} + 15 a + 4\right)\cdot 17 + \left(15 a^{2} + 14 a + 14\right)\cdot 17^{2} + \left(a^{2} + 4 a + 9\right)\cdot 17^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 17^{4} + \left(4 a^{2} + 7 a + 2\right)\cdot 17^{5} + \left(7 a^{2} + 13 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 6 }$ $=$ $4 a^{2} + 9 a + 11 + \left(11 a^{2} + 4 a + 2\right)\cdot 17 + \left(11 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(10 a^{2} + 10 a + 4\right)\cdot 17^{3} + \left(3 a^{2} + 4 a + 10\right)\cdot 17^{4} + \left(5 a^{2} + 10 a + 8\right)\cdot 17^{5} + \left(9 a^{2} + 9 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 7 }$ $=$ $3 a + 9 + \left(14 a^{2} + 14\right)\cdot 17 + \left(11 a^{2} + 16 a + 10\right)\cdot 17^{2} + \left(2 a^{2} + 11 a + 2\right)\cdot 17^{3} + \left(2 a^{2} + 3 a + 5\right)\cdot 17^{4} + \left(10 a^{2} + 16 a + 15\right)\cdot 17^{5} + \left(14 a + 4\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$

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

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

### Character values on conjugacy classes

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