# Properties

 Label 1.113.7t1.a.d Dimension 1 Group $C_7$ Conductor $113$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_7$ Conductor: $113$ Artin number field: Splitting field of 7.7.2081951752609.1 defined by $f= x^{7} - x^{6} - 48 x^{5} - 37 x^{4} + 312 x^{3} + 12 x^{2} - 49 x + 1$ over $\Q$ Size of Galois orbit: 6 Smallest containing permutation representation: $C_7$ Parity: Even Corresponding Dirichlet character: $$\chi_{113}(30,\cdot)$$ 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 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{7} + 4 x + 9$
Roots:
 $r_{ 1 }$ $=$ $2 a^{6} + 7 a^{5} + 4 a^{4} + 3 a^{3} + 9 a^{2} + 3 a + 7 + \left(3 a^{6} + 8 a^{5} + a^{4} + 6 a^{3} + 2 a^{2} + 4 a + 4\right)\cdot 11 + \left(3 a^{6} + 10 a^{5} + 5 a^{4} + 8 a^{3} + 6 a + 3\right)\cdot 11^{2} + \left(9 a^{6} + 7 a^{5} + a^{4} + 10 a^{3} + 3 a^{2} + 8\right)\cdot 11^{3} + \left(8 a^{6} + 2 a^{5} + 4 a^{4} + 9 a^{2} + a + 6\right)\cdot 11^{4} + \left(5 a^{6} + 9 a^{5} + 2 a^{4} + 6 a^{3} + 7 a^{2} + 7 a + 2\right)\cdot 11^{5} + \left(5 a^{6} + 2 a^{4} + 5 a^{3} + a^{2} + 5 a + 3\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 2 }$ $=$ $2 a^{6} + 7 a^{5} + 6 a^{4} + 7 a^{2} + 8 a + 7 + \left(7 a^{5} + 2 a^{4} + 5 a^{3} + 10 a^{2} + 9 a\right)\cdot 11 + \left(7 a^{6} + 3 a^{5} + 4 a^{4} + 6 a^{3} + 3 a^{2} + 5 a + 2\right)\cdot 11^{2} + \left(5 a^{6} + 10 a^{4} + 2 a^{3} + 8 a^{2} + 10 a + 5\right)\cdot 11^{3} + \left(5 a^{6} + 4 a^{4} + 8 a^{3} + 2 a^{2} + 6 a + 9\right)\cdot 11^{4} + \left(4 a^{6} + 6 a^{5} + 9 a^{4} + 4 a^{3} + 10 a^{2} + 9 a + 2\right)\cdot 11^{5} + \left(5 a^{6} + 4 a^{5} + a^{4} + 3 a^{3} + 4 a^{2} + 6 a + 1\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 3 }$ $=$ $2 a^{6} + 10 a^{5} + 4 a^{4} + 6 a^{3} + 8 a^{2} + 5 a + 7 + \left(9 a^{6} + 9 a^{5} + 9 a^{4} + a^{3} + 10 a + 1\right)\cdot 11 + \left(2 a^{6} + 3 a^{5} + 2 a^{4} + 8 a^{3} + a^{2} + 8\right)\cdot 11^{2} + \left(6 a^{6} + 8 a^{5} + 4 a^{4} + 6 a^{3} + a^{2} + 10 a + 5\right)\cdot 11^{3} + \left(2 a^{6} + a^{5} + 9 a^{4} + 7 a^{2} + 6 a + 2\right)\cdot 11^{4} + \left(a^{6} + 9 a^{5} + 5 a^{4} + a^{3} + 9 a^{2} + 4\right)\cdot 11^{5} + \left(4 a^{6} + 5 a^{5} + a^{4} + 10 a^{3} + 6 a^{2} + 2 a + 9\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 4 }$ $=$ $4 a^{6} + 8 a^{5} + 4 a^{4} + 10 a^{3} + 8 a^{2} + 4 a + 6 + \left(9 a^{6} + 3 a^{5} + 10 a^{4} + 3 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 11 + \left(6 a^{6} + 2 a^{4} + 6 a^{3} + 5 a^{2} + 4 a + 7\right)\cdot 11^{2} + \left(4 a^{6} + 9 a^{5} + a^{4} + 6 a^{3} + 3 a^{2} + 1\right)\cdot 11^{3} + \left(3 a^{6} + 3 a^{5} + 6 a^{4} + 7 a^{3} + 8 a^{2} + 3 a + 10\right)\cdot 11^{4} + \left(10 a^{6} + 4 a^{5} + 10 a^{4} + 2 a^{3} + 10 a^{2} + 6 a + 3\right)\cdot 11^{5} + \left(7 a^{6} + 3 a^{5} + 8 a^{4} + 3 a^{3} + 9 a^{2} + 2 a + 8\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 5 }$ $=$ $5 a^{6} + 9 a^{5} + 7 a^{4} + 10 a^{3} + 4 a^{2} + 7 a + \left(7 a^{6} + 9 a^{4} + 5 a^{3} + 6 a^{2} + 8 a + 2\right)\cdot 11 + \left(5 a^{6} + 7 a^{5} + 8 a^{4} + 3 a^{3} + 4 a^{2} + 5 a + 2\right)\cdot 11^{2} + \left(4 a^{6} + 6 a^{5} + 5 a^{4} + 9 a^{3} + 10 a^{2} + 8 a + 9\right)\cdot 11^{3} + \left(7 a^{6} + 5 a^{5} + 5 a^{4} + 9 a^{3} + 9 a^{2} + 1\right)\cdot 11^{4} + \left(3 a^{6} + 5 a^{5} + 8 a^{4} + 4 a^{3} + 7 a + 3\right)\cdot 11^{5} + \left(8 a^{6} + a^{5} + 8 a^{4} + 7 a^{3} + 3 a^{2} + 9 a\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 6 }$ $=$ $9 a^{6} + 4 a^{5} + 7 a^{4} + 7 a^{3} + 8 a^{2} + 2 a + 9 + \left(9 a^{5} + 6 a^{4} + 5 a^{2} + a + 2\right)\cdot 11 + \left(3 a^{6} + 8 a^{5} + 3 a^{4} + 5 a^{3} + 5 a^{2} + 7 a + 4\right)\cdot 11^{2} + \left(9 a^{6} + 7 a^{5} + 4 a^{4} + a^{3} + 4 a^{2} + 4 a\right)\cdot 11^{3} + \left(3 a^{6} + 2 a^{5} + 6 a^{4} + 10 a^{3} + 5 a^{2} + 4 a + 10\right)\cdot 11^{4} + \left(8 a^{6} + 10 a^{5} + a^{3} + 8 a^{2} + 10 a + 1\right)\cdot 11^{5} + \left(9 a^{6} + 3 a^{4} + 9 a^{3} + 2 a^{2} + 2 a + 5\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 7 }$ $=$ $9 a^{6} + 10 a^{5} + a^{4} + 8 a^{3} + 4 a + 9 + \left(2 a^{6} + 3 a^{5} + 4 a^{4} + 9 a^{3} + 3 a^{2} + 4 a + 1\right)\cdot 11 + \left(4 a^{6} + 9 a^{5} + 5 a^{4} + 5 a^{3} + a^{2} + 2 a + 5\right)\cdot 11^{2} + \left(4 a^{6} + 3 a^{5} + 5 a^{4} + 6 a^{3} + 2 a^{2} + 9 a + 2\right)\cdot 11^{3} + \left(a^{6} + 5 a^{5} + 7 a^{4} + 6 a^{3} + a^{2} + 9 a + 3\right)\cdot 11^{4} + \left(10 a^{6} + 10 a^{5} + 6 a^{4} + 7 a^{2} + 2 a + 3\right)\cdot 11^{5} + \left(2 a^{6} + 4 a^{5} + 6 a^{4} + 5 a^{3} + 3 a^{2} + 3 a + 5\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$

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

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

### Character values on conjugacy classes

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