# Properties

 Label 2.16.al_ci 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: $( 1 - 7 x + 16 x^{2} )( 1 - 4 x + 16 x^{2} )$ Frobenius angles: $\pm0.160861246510$, $\pm0.333333333333$ Angle rank: $1$ (numerical) Jacobians: 2

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 130 65520 17280250 4326547680 1100192538250 281474121474000 72061311642105370 18447374928131997120 4722408399252751470250 1208926200925625332878000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 256 4218 66016 1049226 16777168 268449306 4295114176 68720086698 1099511974576

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ah $\times$ 1.16.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ah $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.ah : $$\Q(\sqrt{-15})$$. 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.ad_e $2$ 2.256.ab_jg 2.16.d_e $2$ 2.256.ab_jg 2.16.l_ci $2$ 2.256.ab_jg 2.16.b_ay $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.ad_e $2$ 2.256.ab_jg 2.16.d_e $2$ 2.256.ab_jg 2.16.l_ci $2$ 2.256.ab_jg 2.16.b_ay $3$ (not in LMFDB) 2.16.ap_dk $6$ (not in LMFDB) 2.16.ab_ay $6$ (not in LMFDB) 2.16.p_dk $6$ (not in LMFDB) 2.16.ah_bg $12$ (not in LMFDB) 2.16.h_bg $12$ (not in LMFDB)