# Properties

 Label 2.16.ah_bg 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: $( 1 - 7 x + 16 x^{2} )( 1 + 16 x^{2} )$ Frobenius angles: $\pm0.160861246510$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 4

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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 170 69360 16756730 4276044000 1101267994250 281749132812240 72065710183970330 18446530487208984000 4722372370410814656170 1208927353846394074254000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 272 4090 65248 1050250 16793552 268465690 4294917568 68719562410 1099513023152

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ah $\times$ 1.16.a 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 $\F_{2^{8}}$ is 1.256.ar $\times$ 1.256.bg. The endomorphism algebra for each factor is: 1.256.ar : $$\Q(\sqrt{-15})$$. 1.256.bg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{8}}$.

## 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.h_bg $2$ 2.256.p_abg 2.16.ap_dk $4$ (not in LMFDB) 2.16.ab_ay $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.h_bg $2$ 2.256.p_abg 2.16.ap_dk $4$ (not in LMFDB) 2.16.ab_ay $4$ (not in LMFDB) 2.16.b_ay $4$ (not in LMFDB) 2.16.p_dk $4$ (not in LMFDB) 2.16.al_ci $12$ (not in LMFDB) 2.16.ad_e $12$ (not in LMFDB) 2.16.d_e $12$ (not in LMFDB) 2.16.l_ci $12$ (not in LMFDB)