# Properties

 Label 7.3e4_5e4_11e6.8t37.1c1 Dimension 7 Group $\GL(3,2)$ Conductor $3^{4} \cdot 5^{4} \cdot 11^{6}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $7$ Group: $\GL(3,2)$ Conductor: $89685275625= 3^{4} \cdot 5^{4} \cdot 11^{6}$ Artin number field: Splitting field of $f= x^{7} - x^{6} - 9 x^{5} + 21 x^{4} - 3 x^{3} - 23 x^{2} + 10 x + 9$ over $\Q$ Size of Galois orbit: 1 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 $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{3} + 2 x + 11$
Roots:
 $r_{ 1 }$ $=$ $5 a^{2} + 4 + \left(8 a^{2} + 4 a + 4\right)\cdot 13 + \left(2 a^{2} + 7 a + 5\right)\cdot 13^{2} + \left(7 a^{2} + 11 a + 9\right)\cdot 13^{3} + \left(9 a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(6 a^{2} + 4 a + 5\right)\cdot 13^{5} + \left(12 a^{2} + 1\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ $r_{ 2 }$ $=$ $8 + 3\cdot 13 + 5\cdot 13^{2} + 9\cdot 13^{3} + 7\cdot 13^{4} + 12\cdot 13^{5} + 8\cdot 13^{6} +O\left(13^{ 7 }\right)$ $r_{ 3 }$ $=$ $4 a + 9 + \left(a^{2} + 11 a + 2\right)\cdot 13 + \left(a + 5\right)\cdot 13^{2} + \left(5 a^{2} + 2 a + 12\right)\cdot 13^{3} + \left(4 a^{2} + 9 a + 12\right)\cdot 13^{4} + \left(5 a^{2} + 12 a + 5\right)\cdot 13^{5} + \left(11 a^{2} + a + 1\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ $r_{ 4 }$ $=$ $7 a^{2} + 2 a + 1 + \left(12 a^{2} + 9 a + 5\right)\cdot 13 + \left(11 a^{2} + 12\right)\cdot 13^{2} + \left(5 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(6 a^{2} + 2 a + 11\right)\cdot 13^{4} + \left(11 a^{2} + a + 9\right)\cdot 13^{5} + \left(6 a^{2} + 6 a + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ $r_{ 5 }$ $=$ $12 a^{2} + 8 a + 9 + \left(9 a^{2} + 8 a + 10\right)\cdot 13 + \left(5 a^{2} + 5 a\right)\cdot 13^{2} + \left(11 a^{2} + 5 a + 2\right)\cdot 13^{3} + \left(8 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(a^{2} + 8 a + 12\right)\cdot 13^{5} + \left(9 a^{2} + 4 a + 9\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ $r_{ 6 }$ $=$ $6 a^{2} + 7 a + 4 + \left(12 a^{2} + 5 a + 9\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(2 a^{2} + 11 a + 12\right)\cdot 13^{3} + \left(2 a^{2} + 9\right)\cdot 13^{4} + \left(9 a^{2} + 12 a + 6\right)\cdot 13^{5} + \left(7 a^{2} + 4 a + 9\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ $r_{ 7 }$ $=$ $9 a^{2} + 5 a + 5 + \left(7 a^{2} + 3\right)\cdot 13 + \left(4 a^{2} + 12\right)\cdot 13^{2} + \left(7 a^{2} + 9 a\right)\cdot 13^{3} + \left(7 a^{2} + 2 a + 9\right)\cdot 13^{4} + \left(4 a^{2} + 11\right)\cdot 13^{5} + \left(4 a^{2} + 8 a + 7\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$

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

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

### Character values on conjugacy classes

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