# Properties

 Label 3.614656.42t37.a.a Dimension $3$ Group $\GL(3,2)$ Conductor $614656$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $$614656$$$$\medspace = 2^{8} \cdot 7^{4}$$ Artin number field: Galois closure of 7.3.1475789056.2 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 7.3.1475789056.2

## Defining polynomial

 $f(x)$ $=$ $x^{7} - 7 x^{5} - 14 x^{4} - 7 x^{3} - 7 x + 2$.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{3} + 2 x + 11$

Roots:
 $r_{ 1 }$ $=$ $8 a^{2} + 6 a + 10 + \left(5 a^{2} + 10 a + 5\right)\cdot 13 + \left(7 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(5 a^{2} + 3 a + 4\right)\cdot 13^{3} + \left(2 a^{2} + a + 3\right)\cdot 13^{4} + \left(2 a^{2} + 5 a + 8\right)\cdot 13^{5} + \left(8 a^{2} + 5 a + 1\right)\cdot 13^{6} + \left(4 a + 9\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 2 }$ $=$ $12 a + 8 + \left(7 a^{2} + 7 a + 7\right)\cdot 13 + \left(10 a^{2} + 2 a + 8\right)\cdot 13^{2} + \left(4 a^{2} + 7\right)\cdot 13^{3} + 12 a\cdot 13^{4} + \left(6 a^{2} + 9\right)\cdot 13^{5} + \left(9 a^{2} + 7 a + 7\right)\cdot 13^{6} + 2 a\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 3 }$ $=$ $5 + 6\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 6\cdot 13^{5} + 3\cdot 13^{6} + 7\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 4 }$ $=$ $9 a^{2} + 5 a + 11 + \left(6 a^{2} + a + 12\right)\cdot 13 + \left(a^{2} + 6 a + 3\right)\cdot 13^{2} + \left(9 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(2 a^{2} + 7 a + 3\right)\cdot 13^{4} + \left(3 a^{2} + 10 a + 1\right)\cdot 13^{5} + \left(9 a^{2} + 3 a + 3\right)\cdot 13^{6} + \left(7 a^{2} + 8 a + 8\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 5 }$ $=$ $12 a^{2} + 8 a + 2 + \left(10 a^{2} + 12 a + 1\right)\cdot 13 + \left(2 a^{2} + 9 a + 10\right)\cdot 13^{2} + \left(3 a + 9\right)\cdot 13^{3} + \left(12 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(10 a^{2} + 9 a + 8\right)\cdot 13^{6} + \left(11 a + 7\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 6 }$ $=$ $5 a^{2} + 10 + \left(8 a^{2} + 12 a + 10\right)\cdot 13 + \left(8 a^{2} + 9 a + 4\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 1\right)\cdot 13^{3} + \left(11 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(7 a^{2} + 11 a + 7\right)\cdot 13^{5} + \left(6 a^{2} + 12 a + 12\right)\cdot 13^{6} + \left(4 a^{2} + 5 a + 3\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 7 }$ $=$ $5 a^{2} + 8 a + 6 + \left(7 a + 7\right)\cdot 13 + \left(8 a^{2} + a + 9\right)\cdot 13^{2} + \left(2 a^{2} + 9 a + 4\right)\cdot 13^{3} + \left(10 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(4 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(8 a^{2} + 1\right)\cdot 13^{6} + \left(11 a^{2} + 6 a + 2\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.