# Properties

 Label 3.14462809.42t37.a.a Dimension $3$ Group $\GL(3,2)$ Conductor $14462809$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $$14462809$$$$\medspace = 3803^{2}$$ Artin number field: Galois closure of 7.3.14462809.1 Galois orbit size: $2$ Smallest permutation container: $\PSL(2,7)$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $\PSL(2,7)$ Projective field: Galois closure of 7.3.14462809.1

## Defining polynomial

 $f(x)$ $=$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 13 x^{3} + 17 x^{2} - 10 x + 1$.

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:
 $r_{ 1 }$ $=$ $4 + 8\cdot 11 + 4\cdot 11^{2} + 11^{3} + 8\cdot 11^{4} + 9\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 2 }$ $=$ $3 a^{2} + 6 a + 9 + \left(a + 5\right)\cdot 11 + \left(8 a^{2} + 7 a + 6\right)\cdot 11^{2} + \left(3 a^{2} + 3 a + 10\right)\cdot 11^{3} + \left(8 a^{2} + 5 a + 9\right)\cdot 11^{4} + \left(8 a^{2} + 6 a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 3 }$ $=$ $8 a + 5 + \left(5 a^{2} + 2 a + 8\right)\cdot 11 + \left(10 a^{2} + 6 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 10 a + 3\right)\cdot 11^{3} + \left(8 a^{2} + a + 10\right)\cdot 11^{4} + \left(5 a^{2} + 5\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 4 }$ $=$ $5 a^{2} + 2 a + 5 + \left(6 a^{2} + 7 a\right)\cdot 11 + \left(5 a^{2} + a + 10\right)\cdot 11^{2} + \left(7 a^{2} + 3\right)\cdot 11^{3} + \left(10 a^{2} + 9 a + 5\right)\cdot 11^{4} + \left(3 a^{2} + 7\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 5 }$ $=$ $3 a^{2} + 9 a + 6 + \left(6 a^{2} + 8 a + 7\right)\cdot 11 + \left(4 a^{2} + 8\right)\cdot 11^{2} + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{3} + \left(3 a^{2} + 7 a + 3\right)\cdot 11^{4} + \left(7 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 6 }$ $=$ $3 a^{2} + 6 + \left(9 a^{2} + 6 a\right)\cdot 11 + 8 a\cdot 11^{2} + \left(4 a^{2} + a + 3\right)\cdot 11^{3} + \left(7 a^{2} + 5 a + 8\right)\cdot 11^{4} + \left(6 a^{2} + 2 a + 3\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 7 }$ $=$ $8 a^{2} + 8 a + 1 + \left(5 a^{2} + 6 a + 2\right)\cdot 11 + \left(3 a^{2} + 8 a + 4\right)\cdot 11^{2} + \left(7 a + 2\right)\cdot 11^{3} + \left(5 a^{2} + 3 a + 9\right)\cdot 11^{4} + \left(7 a^{2} + 4 a\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $3$ $21$ $2$ $(1,4)(2,3)$ $-1$ $56$ $3$ $(1,6,3)(2,5,7)$ $0$ $42$ $4$ $(1,7,6,2)(4,5)$ $1$ $24$ $7$ $(1,5,4,7,6,2,3)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $24$ $7$ $(1,7,3,4,2,5,6)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$

The blue line marks the conjugacy class containing complex conjugation.