# Properties

 Label 1.37.9t1.a.f Dimension $1$ Group $C_9$ Conductor $37$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_9$ Conductor: $$37$$ Artin number field: Galois closure of 9.9.3512479453921.1 Galois orbit size: $6$ Smallest permutation container: $C_9$ Parity: even Dirichlet character: $$\chi_{37}(7,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{9} - x^{8} - 16 x^{7} + 11 x^{6} + 66 x^{5} - 32 x^{4} - 73 x^{3} + 7 x^{2} + 7 x - 1$.

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

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,9)(3,4,6)(5,8,7)$ $-\zeta_{9}^{3} - 1$ $1$ $3$ $(1,9,2)(3,6,4)(5,7,8)$ $\zeta_{9}^{3}$ $1$ $9$ $(1,6,8,2,3,7,9,4,5)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $1$ $9$ $(1,8,3,9,5,6,2,7,4)$ $-\zeta_{9}^{4} - \zeta_{9}$ $1$ $9$ $(1,3,5,2,4,8,9,6,7)$ $\zeta_{9}^{5}$ $1$ $9$ $(1,7,6,9,8,4,2,5,3)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,4,7,2,6,5,9,3,8)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,5,4,9,7,3,2,8,6)$ $\zeta_{9}$

The blue line marks the conjugacy class containing complex conjugation.