# Properties

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

# Learn more about

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 11 x + 57 x^{2} - 176 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.0728689886706$, $\pm0.368631800070$ Angle rank: $2$ (numerical) Number field: 4.0.78057.3 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 127 63627 16863568 4277325075 1096735474927 281339964014400 72060071974132327 18447444892188384675 4722402949892095217488 1208926063435230833771787

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 250 4119 65266 1045926 16769167 268444686 4295130466 68720007399 1099511849530

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.78057.3.
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.l_cf $2$ 2.256.ah_aeh