# Properties

 Label 2.16.ah_bh 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: $1 - 7 x + 33 x^{2} - 112 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.172472086823$, $\pm0.494194579844$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Galois group: $C_2^2$ Jacobians: 29

This isogeny class is simple but not 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 29 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 171 69939 16842816 4280336739 1101267647451 281748310329600 72065710292698251 18446394710506428099 4722354708206963052096 1208926591225993760495859

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 274 4111 65314 1050250 16793503 268465690 4294885954 68719305391 1099512329554

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{12}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

 Subfield Primitive Model $\F_{2}$ 2.2.ad_f $\F_{2}$ 2.2.d_f

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.h_bh $2$ 2.256.r_bh 2.16.o_dd $3$ (not in LMFDB) 2.16.ao_dd $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.h_bh $2$ 2.256.r_bh 2.16.o_dd $3$ (not in LMFDB) 2.16.ao_dd $6$ (not in LMFDB) 2.16.a_ar $6$ (not in LMFDB) 2.16.a_r $12$ (not in LMFDB)