# Properties

 Label 2.16.aj_bw Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 9 x + 48 x^{2} - 144 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.193865395619$, $\pm0.401408407366$ Angle rank: $2$ (numerical) Number field: 4.0.21964.1 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically simple.

## Newton polygon

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

## Point counts

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

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 152 69616 17341832 4309648096 1099500113912 281555775976144 72068686444863848 18447097843715097024 4722328938530828954072 1208921252882057995001776

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 272 4232 65760 1048568 16782032 268476776 4295049664 68718930392 1099507474352

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.21964.1.
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_bw $2$ 2.256.p_iq