# Properties

 Label 2.1859.7t2.a Dimension $2$ Group $D_{7}$ Conductor $1859$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{7}$ Conductor: $$1859$$$$\medspace = 11 \cdot 13^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 7.1.6424482779.1 Galois orbit size: $3$ Smallest permutation container: $D_{7}$ Parity: odd Projective image: $D_7$ Projective field: Galois closure of 7.1.6424482779.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$15 a + 9 + \left(13 a + 14\right)\cdot 17 + 5\cdot 17^{2} + \left(5 a + 13\right)\cdot 17^{3} + \left(a + 13\right)\cdot 17^{4} +O(17^{5})$$ 15*a + 9 + (13*a + 14)*17 + 5*17^2 + (5*a + 13)*17^3 + (a + 13)*17^4+O(17^5) $r_{ 2 }$ $=$ $$15 a + 10 + 14 a\cdot 17 + 15 a\cdot 17^{2} + \left(12 a + 16\right)\cdot 17^{3} + \left(a + 13\right)\cdot 17^{4} +O(17^{5})$$ 15*a + 10 + 14*a*17 + 15*a*17^2 + (12*a + 16)*17^3 + (a + 13)*17^4+O(17^5) $r_{ 3 }$ $=$ $$15 + 3\cdot 17 + 4\cdot 17^{2} + 7\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})$$ 15 + 3*17 + 4*17^2 + 7*17^3 + 2*17^4+O(17^5) $r_{ 4 }$ $=$ $$2 a + 8 + 2 a\cdot 17 + \left(a + 1\right)\cdot 17^{2} + \left(4 a + 13\right)\cdot 17^{3} + \left(15 a + 2\right)\cdot 17^{4} +O(17^{5})$$ 2*a + 8 + 2*a*17 + (a + 1)*17^2 + (4*a + 13)*17^3 + (15*a + 2)*17^4+O(17^5) $r_{ 5 }$ $=$ $$11 a + 14 + \left(11 a + 8\right)\cdot 17 + 3\cdot 17^{2} + \left(11 a + 12\right)\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O(17^{5})$$ 11*a + 14 + (11*a + 8)*17 + 3*17^2 + (11*a + 12)*17^3 + (12*a + 11)*17^4+O(17^5) $r_{ 6 }$ $=$ $$2 a + 7 + \left(3 a + 13\right)\cdot 17 + \left(16 a + 9\right)\cdot 17^{2} + 11 a\cdot 17^{3} + \left(15 a + 10\right)\cdot 17^{4} +O(17^{5})$$ 2*a + 7 + (3*a + 13)*17 + (16*a + 9)*17^2 + 11*a*17^3 + (15*a + 10)*17^4+O(17^5) $r_{ 7 }$ $=$ $$6 a + 8 + \left(5 a + 9\right)\cdot 17 + \left(16 a + 9\right)\cdot 17^{2} + \left(5 a + 5\right)\cdot 17^{3} + \left(4 a + 13\right)\cdot 17^{4} +O(17^{5})$$ 6*a + 8 + (5*a + 9)*17 + (16*a + 9)*17^2 + (5*a + 5)*17^3 + (4*a + 13)*17^4+O(17^5)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $c3$ $1$ $1$ $()$ $2$ $2$ $2$ $7$ $2$ $(1,5)(3,6)(4,7)$ $0$ $0$ $0$ $2$ $7$ $(1,2,5,3,7,4,6)$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $2$ $7$ $(1,5,7,6,2,3,4)$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $2$ $7$ $(1,3,6,5,4,2,7)$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$
The blue line marks the conjugacy class containing complex conjugation.