# Properties

 Label 1.71.7t1.a Dimension $1$ Group $C_7$ Conductor $71$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_7$ Conductor: $$71$$ Artin number field: Galois closure of 7.7.128100283921.1 Galois orbit size: $6$ Smallest permutation container: $C_7$ Parity: even Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $c3$ $c4$ $c5$ $c6$ $1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $7$ $(1,2,7,5,3,4,6)$ $\zeta_{7}$ $\zeta_{7}^{2}$ $\zeta_{7}^{3}$ $\zeta_{7}^{4}$ $\zeta_{7}^{5}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $1$ $7$ $(1,7,3,6,2,5,4)$ $\zeta_{7}^{2}$ $\zeta_{7}^{4}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}$ $\zeta_{7}^{3}$ $\zeta_{7}^{5}$ $1$ $7$ $(1,5,6,7,4,2,3)$ $\zeta_{7}^{3}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{2}$ $\zeta_{7}^{5}$ $\zeta_{7}$ $\zeta_{7}^{4}$ $1$ $7$ $(1,3,2,4,7,6,5)$ $\zeta_{7}^{4}$ $\zeta_{7}$ $\zeta_{7}^{5}$ $\zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{3}$ $1$ $7$ $(1,4,5,2,6,3,7)$ $\zeta_{7}^{5}$ $\zeta_{7}^{3}$ $\zeta_{7}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{4}$ $\zeta_{7}^{2}$ $1$ $7$ $(1,6,4,3,5,7,2)$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{5}$ $\zeta_{7}^{4}$ $\zeta_{7}^{3}$ $\zeta_{7}^{2}$ $\zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.