# Properties

 Label 2.17.am_cp Base Field $\F_{17}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{17}$ Dimension: $2$ L-polynomial: $1 - 12 x + 67 x^{2} - 204 x^{3} + 289 x^{4}$ Frobenius angles: $\pm0.112997812760$, $\pm0.326838618982$ Angle rank: $2$ (numerical) Number field: 4.0.58896.2 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=9x^6+15x^5+2x^4+9x^2+16x+14$
• $y^2=5x^6+15x^5+3x^4+2x^3+x^2+8x+10$
• $y^2=10x^6+5x^5+14x^4+15x^3+14x^2+14x+6$
• $y^2=6x^6+x^5+10x^4+2x^3+9x^2+4x+5$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 141 80793 24491700 6996108249 2015297698821 582512650102800 168380717002045269 48662715418390124649 14063246313837597011700 4064240468448309770809353

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 280 4986 83764 1419366 24133030 410345718 6975975844 118589241402 2015998395400

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
 The endomorphism algebra of this simple isogeny class is 4.0.58896.2.
All geometric endomorphisms are defined over $\F_{17}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.17.m_cp $2$ (not in LMFDB)