# Properties

 Label 1.19.9t1.a.b Dimension 1 Group $C_9$ Conductor $19$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $1$ Group: $C_9$ Conductor: $19$ Artin number field: Splitting field of $$\Q(\zeta_{19})^+$$ defined by $f= x^{9} - x^{8} - 8 x^{7} + 7 x^{6} + 21 x^{5} - 15 x^{4} - 20 x^{3} + 10 x^{2} + 5 x - 1$ over $\Q$ Size of Galois orbit: 6 Smallest containing permutation representation: $C_9$ Parity: Even Corresponding Dirichlet character: $$\chi_{19}(5,\cdot)$$ 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 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{3} + 2 x + 9$
Roots:
 $r_{ 1 }$ $=$ $3 a^{2} + 4 a + \left(a^{2} + 6 a + 2\right)\cdot 11 + \left(4 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(a^{2} + 3 a + 5\right)\cdot 11^{3} + \left(4 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(8 a^{2} + 8 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 2 }$ $=$ $4 a^{2} + 2 a + 8 + \left(2 a^{2} + 4 a + 9\right)\cdot 11 + \left(5 a^{2} + 9 a + 9\right)\cdot 11^{2} + \left(9 a^{2} + a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{4} + \left(9 a^{2} + 6 a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 3 }$ $=$ $2 a^{2} + 8 a + 9 + \left(7 a^{2} + 5 a + 8\right)\cdot 11 + \left(a^{2} + 8 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 4 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 4 a + 7\right)\cdot 11^{4} + \left(2 a^{2} + 7 a + 7\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 4 }$ $=$ $9 a^{2} + 6 a + 8 + \left(a^{2} + 2 a + 2\right)\cdot 11 + \left(5 a^{2} + 9\right)\cdot 11^{2} + \left(6 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(6 a^{2} + 9 a + 6\right)\cdot 11^{4} + \left(2 a^{2} + 8 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 5 }$ $=$ $5 a^{2} + a + 2 + \left(a^{2} + a + 1\right)\cdot 11 + \left(4 a^{2} + 4 a + 1\right)\cdot 11^{2} + \left(3 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 6 }$ $=$ $10 a^{2} + a + 2 + \left(7 a^{2} + 2 a + 7\right)\cdot 11 + \left(a^{2} + 7 a + 4\right)\cdot 11^{2} + \left(3 a^{2} + 4 a\right)\cdot 11^{3} + \left(3 a + 9\right)\cdot 11^{4} + \left(4 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 7 }$ $=$ $7 a^{2} + \left(7 a^{2} + 8 a + 7\right)\cdot 11 + \left(4 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(5 a^{2} + 9 a + 2\right)\cdot 11^{3} + \left(6 a^{2} + a + 1\right)\cdot 11^{4} + \left(5 a^{2} + 3 a\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 8 }$ $=$ $7 a^{2} + 10 a + \left(6 a^{2} + 8 a + 2\right)\cdot 11 + \left(a^{2} + 9 a + 4\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} + \left(10 a^{2} + 10 a + 9\right)\cdot 11^{4} + \left(8 a^{2} + a\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ $r_{ 9 }$ $=$ $8 a^{2} + a + 5 + \left(7 a^{2} + 5 a + 3\right)\cdot 11 + \left(4 a^{2} + 9 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 7 a + 9\right)\cdot 11^{3} + \left(5 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(7 a^{2} + 5 a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(1,2,8,6,3,9,4,5,7)$ $(1,4,6)(2,5,3)(7,9,8)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,6,4)(2,3,5)(7,8,9)$ $-\zeta_{9}^{3} - 1$ $1$ $3$ $(1,4,6)(2,5,3)(7,9,8)$ $\zeta_{9}^{3}$ $1$ $9$ $(1,2,8,6,3,9,4,5,7)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,8,3,4,7,2,6,9,5)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,3,7,6,5,8,4,2,9)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $1$ $9$ $(1,9,2,4,8,5,6,7,3)$ $\zeta_{9}$ $1$ $9$ $(1,5,9,6,2,7,4,3,8)$ $\zeta_{9}^{5}$ $1$ $9$ $(1,7,5,4,9,3,6,8,2)$ $-\zeta_{9}^{4} - \zeta_{9}$
The blue line marks the conjugacy class containing complex conjugation.