# Properties

 Label 2.16.aj_bv Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 9 x + 47 x^{2} - 144 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.177258786107$, $\pm0.410961538347$ Angle rank: $2$ (numerical) Number field: 4.0.466137.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2)x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^3+a^2+a)x^3+(a^3+a+1)x^2+x+a+1$
• $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+a)x^6+(a^3+a)x^5+(a^3+a)x^4+(a^3+1)x^3+x^2+x+a^3+a^2+a$
• $y^2+(x^3+ax+a)y=a^3x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a^3+a+1)x^3+(a^2+a)x^2+x+a^3+a$
• $y^2+(x^3+(a+1)x+a+1)y=(a^3+1)x^6+a^3x^5+a^3x^4+(a^3+a^2+1)x^3+(a^3+a+1)x^2+x+1$
• $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+1)x^6+(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+(a^3+a^2+a)x^3+a^2x^2+a^3x+a^3+a^2+1$
• $y^2+(x^3+ax+a)y=a^3x^6+(a^3+1)x^5+(a^3+1)x^4+(a^3+a^2+1)x^3+(a^2+a)x^2+(a^3+a)x+a^3+a^2+1$
• $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+a)x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+(a^3+1)x^3+(a^3+a^2+a+1)x^2+(a^3+a^2+a+1)x+1$
• $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2+1)x^6+(a^3+a^2+a)x^5+(a^3+a^2+a)x^4+(a^3+a+1)x^3+(a^3+a^2)x^2+(a^3+a^2)x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 151 69007 17229100 4300861275 1099406044381 281594960306800 72072136714963291 18447235639717043475 4722334799802285208900 1208921907176053523048887

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 270 4205 65626 1048478 16784367 268489628 4295081746 68719015685 1099508069430

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.466137.1.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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 2.16.j_bv $2$ 2.256.n_ez