Properties

 Label 2.143.5t2.a.b Dimension $2$ Group $D_{5}$ Conductor $143$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $$143$$$$\medspace = 11 \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.1.20449.1 Galois orbit size: $2$ Smallest permutation container: $D_{5}$ Parity: odd Determinant: 1.143.2t1.a.a Projective image: $D_5$ Projective stem field: 5.1.20449.1

Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} - x^{3} + 3 x - 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$13 a + 3 + \left(2 a + 11\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(8 a + 9\right)\cdot 17^{3} + \left(13 a + 16\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$9 a + 7 + \left(a + 8\right)\cdot 17 + \left(8 a + 5\right)\cdot 17^{2} + \left(a + 16\right)\cdot 17^{3} + \left(11 a + 2\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 3 }$ $=$ $$10 + 12\cdot 17 + 14\cdot 17^{2} + 9\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$8 a + 16 + 15 a\cdot 17 + \left(8 a + 12\right)\cdot 17^{2} + \left(15 a + 9\right)\cdot 17^{3} + \left(5 a + 12\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 5 }$ $=$ $$4 a + 16 + 14 a\cdot 17 + \left(3 a + 6\right)\cdot 17^{2} + \left(8 a + 5\right)\cdot 17^{3} + \left(3 a + 4\right)\cdot 17^{4} +O(17^{5})$$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $2$ $5$ $2$ $(1,3)(2,5)$ $0$ $2$ $5$ $(1,5,2,3,4)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $2$ $5$ $(1,2,4,5,3)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.